Question 12 Marks
Find the square root in decimal from:
0.00002025
0.00002025
Answer
Hence, the square root of 0.00002025 is 0.0045
View full question & answer→Hence, the square root of 0.00002025 is 0.0045Questions
2 Mark Question
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118 questions · self-marked practice — reveal the answer and mark yourself.
Question 22 Marks
Find the squares of the following numbers.
512
512
Answer
View full question & answer→$(512)^2$
Here $a=1, b=2$
$\therefore(5 a b)^2=(250+a b) \times 1000+(a b)^2$
$\therefore(512)^2=(250+12) \times 1000+(12)^2$
$=262 \times 1000+144$
$=262000+144$
$=262144$
Here $a=1, b=2$
$\therefore(5 a b)^2=(250+a b) \times 1000+(a b)^2$
$\therefore(512)^2=(250+12) \times 1000+(12)^2$
$=262 \times 1000+144$
$=262000+144$
$=262144$
Question 32 Marks
Using prime factorization method, find the following numbers are perfect squares?
441
441
Answer
View full question & answer→441 = 3 × 3 × 7 × 7
$\begin{array}{c|c} 3& 441 \\ \hline 3 & 147 \\\hline 7&49 \\\hline 7&7\\\hline&1 \end{array}$
Grouping them into pairs of equal factors,
441 = 3 × 3 × (7 × 7)
There are no left out of pairs. Hence, 441 is a perfect square.
$\begin{array}{c|c} 3& 441 \\ \hline 3 & 147 \\\hline 7&49 \\\hline 7&7\\\hline&1 \end{array}$
Grouping them into pairs of equal factors,
441 = 3 × 3 × (7 × 7)
There are no left out of pairs. Hence, 441 is a perfect square.
Question 42 Marks
Find the length of a side of a square playground whose area is equal to the area of a rectangular field of diamensions 72m and 338m.
Answer
View full question & answer→The area of the playground $=72 \times 338=24336 m^2$
The length of one side of a square is equal to the square root of its area. Hence, we just need to find the square root of 24336.
Hence, the length of one side of the playground is 156 metres.
The length of one side of a square is equal to the square root of its area. Hence, we just need to find the square root of 24336.
Hence, the length of one side of the playground is 156 metres.
Question 52 Marks
Find the least number which be added to the following numbers to make tham a perfect square:
4515600
4515600
Answer
View full question & answer→Using the long division method,
We can see that 4515600 is 25 more than $2125^2$. Hence, we have to add 25 to 4515600 to get a perfect square.
We can see that 4515600 is 25 more than $2125^2$. Hence, we have to add 25 to 4515600 to get a perfect square.
Question 62 Marks
The area of a square playground is 256.6404 square metres. Find the length of one side of the playground.
Answer
View full question & answer→The length of one side of the playground is the square root of its area.
So, the length of one side of the playground is 16.02 metres.
So, the length of one side of the playground is 16.02 metres.
Question 72 Marks
The area of a square field is $30\frac{1}{4}\text{m}^2$ Calculate the length of the side of the square.
Answer
View full question & answer→The length of one side is equal to the square root of the area of the field. Hence, we just need to calculate the value of $\sqrt{30\frac{1}{4}}$
Calculate the value of $\sqrt{30\frac{1}{4}}$
We have,
$\sqrt{30\frac{1}{4}}=\frac{\sqrt{121}}{\sqrt{14}}$
Now, calculating the square root of the numerator and the denominator,
$\sqrt{121}=\sqrt{11\times11}=11$
$\sqrt{4}=2$
Therefore, the length of the side of the square $\sqrt{30\frac{1}{4}}=\frac{11}{2}=5\frac{1}{2}\text{m}$
Calculate the value of $\sqrt{30\frac{1}{4}}$
We have,
$\sqrt{30\frac{1}{4}}=\frac{\sqrt{121}}{\sqrt{14}}$
Now, calculating the square root of the numerator and the denominator,
$\sqrt{121}=\sqrt{11\times11}=11$
$\sqrt{4}=2$
Therefore, the length of the side of the square $\sqrt{30\frac{1}{4}}=\frac{11}{2}=5\frac{1}{2}\text{m}$
Question 82 Marks
Find the squares of the following numbers using the identity $(a-b)^2=a^2-2 a b+b^2$ :
498
498
Answer
View full question & answer→$(a-b)^2=a^2-2 a b+b^2$
$(498)^2=(500-5)^2$
$=(500)^2-2 \times 500 \times 2+(2)^2$
$=250000-2000+4$
$=250004-2000$
$=248004$
$(498)^2=(500-5)^2$
$=(500)^2-2 \times 500 \times 2+(2)^2$
$=250000-2000+4$
$=250004-2000$
$=248004$
Question 92 Marks
Find the square root of:
$75\frac{46}{49}$
$75\frac{46}{49}$
Answer
View full question & answer→We know, $\sqrt{75\frac{46}{49}}=\sqrt{\frac{3721}{49}}=\frac{\sqrt{3721}}{\sqrt{49}}$ Now, let us compute the square roots of the numerator and the denominator separately.
$\sqrt{49}=7$ $\therefore\sqrt{75\frac{46}{49}}=\frac{61}{7}$
$\sqrt{49}=7$ $\therefore\sqrt{75\frac{46}{49}}=\frac{61}{7}$
Question 102 Marks
Find the square root of the following by long division method:4008004
Answer
Hence, the square root of 4008004 is 2002
View full question & answer→Hence, the square root of 4008004 is 2002Question 112 Marks
Find the squares of the following numbers.
$95$
$95$
Answer
View full question & answer→$(95)^2$
$\text { Here } n=9$
$\therefore n(n+1)=9(9+1)$
$=9 \times 10=90$
$\therefore(95)^2=9025$
$\text { Here } n=9$
$\therefore n(n+1)=9(9+1)$
$=9 \times 10=90$
$\therefore(95)^2=9025$
Question 122 Marks
Find the square root of:
$3\frac{942}{2209}$
$3\frac{942}{2209}$
Answer
View full question & answer→We know, $\sqrt{3\frac{942}{2209}}=\sqrt{\frac{7569}{2209}}=\frac{\sqrt{7569}}{\sqrt{2209}}$ Now, let us compute the square roots of the numerator and the denominator separately.
$\therefore\sqrt{3\frac{942}{2209}}=\frac{87}{47}$
$\therefore\sqrt{3\frac{942}{2209}}=\frac{87}{47}$
Question 132 Marks
Find the square root of:
$10\frac{151}{225}$
$10\frac{151}{225}$
Answer
View full question & answer→We know,
$\sqrt{10\frac{151}{225}}=\sqrt{\frac{2401}{225}}=\frac{\sqrt{2401}}{\sqrt{225}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$\sqrt{2401}=\sqrt{7\times7\times7\times7}=7\times7=49$
$\sqrt{225}=\sqrt{3\times3\times5\times5}=3\times5=15$
$\therefore\sqrt{10\frac{151}{225}}=\frac{49}{15}=3\frac{4}{15}$
$\sqrt{10\frac{151}{225}}=\sqrt{\frac{2401}{225}}=\frac{\sqrt{2401}}{\sqrt{225}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$\sqrt{2401}=\sqrt{7\times7\times7\times7}=7\times7=49$
$\sqrt{225}=\sqrt{3\times3\times5\times5}=3\times5=15$
$\therefore\sqrt{10\frac{151}{225}}=\frac{49}{15}=3\frac{4}{15}$
Question 142 Marks
Which of the following triplets are pythagorean?
$(18,80,82)$
$(18,80,82)$
Answer
View full question & answer→A triplet $(a, b, c)$ is called Pythagorean if the sum of the squares of the two smallest numbers is equal to the square of the biggest number.
The two smallest numbers are 18 and 80 . The sum of their squares is,
$18^2+80^2=6724=82^2$
Hence, $(18,80,82)$ is a Pythagorean triplet.
The two smallest numbers are 18 and 80 . The sum of their squares is,
$18^2+80^2=6724=82^2$
Hence, $(18,80,82)$ is a Pythagorean triplet.
Question 152 Marks
Find the squares of the following numbers using the identity $(a+b)^2=a^2+2 a b+b^2$ :
510
510
Answer
View full question & answer→$(a+b)^2=a^2+2 a b+b^2$
$(510)^2=(500+10)^2$
$=(500)^2+2 \times 500 \times 10 \times(10)^2$
$=250000+10000+100$
$=260100$
$(510)^2=(500+10)^2$
$=(500)^2+2 \times 500 \times 10 \times(10)^2$
$=250000+10000+100$
$=260100$
Question 162 Marks
Find the square root of the following by long division method:82264900
Answer
Hence, the square root of 82264900 is 9070
View full question & answer→Hence, the square root of 82264900 is 9070Question 172 Marks
Find the squares of the following numbers:
$451$
$451$
Answer
View full question & answer→$(451)^2=(400+51)^2$
$\left\{(a+b)^2=a^2+2 a b+b^2\right\}$
$=(400)^2+2 \times 400 \times 51+(51)^2$
$=160000+4080+2601$
$=203401$
$\left\{(a+b)^2=a^2+2 a b+b^2\right\}$
$=(400)^2+2 \times 400 \times 51+(51)^2$
$=160000+4080+2601$
$=203401$
Question 182 Marks
Which of the following triplets are pythagorean?
$(8,15,17)$
$(8,15,17)$
Answer
View full question & answer→A triplet ( $a, b, c$ ) is called Pythagorean if the sum of the squares of the two smallest numbers is equal to the square of the biggest number. The two smallest numbers are 8 and 15 . The sum of their squares is,
$8^2+15^2=289=17^2$
Hence, $(8,15,17)$ is a Pythagorean triplet.
$8^2+15^2=289=17^2$
Hence, $(8,15,17)$ is a Pythagorean triplet.
Question 192 Marks
Using square root table, find the square root:
25725
25725
Answer
View full question & answer→Using the table to find $\sqrt{3}$ and $\sqrt{7}$$\sqrt{25725}=\sqrt{3\times5\times5\times7\times7\times7}$
$=\sqrt{3}\times5\times7\times\sqrt{7}$
$=1.732\times5\times7\times2.646$
$=160.41$
$=\sqrt{3}\times5\times7\times\sqrt{7}$
$=1.732\times5\times7\times2.646$
$=160.41$
Question 202 Marks
Find the square root of:
$2\frac{137}{196}$
$2\frac{137}{196}$
Answer
View full question & answer→We know,
$\sqrt{2\frac{137}{196}}=\sqrt{\frac{529}{196}}=\frac{\sqrt{529}}{\sqrt{196}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$\sqrt{529}=\sqrt{23\times23}=23$
$\sqrt{196}=\sqrt{2\times2\times7\times7}=2\times7=14$
$\therefore\sqrt{2\frac{137}{196}}=\frac{23}{14}$
$\sqrt{2\frac{137}{196}}=\sqrt{\frac{529}{196}}=\frac{\sqrt{529}}{\sqrt{196}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$\sqrt{529}=\sqrt{23\times23}=23$
$\sqrt{196}=\sqrt{2\times2\times7\times7}=2\times7=14$
$\therefore\sqrt{2\frac{137}{196}}=\frac{23}{14}$
Question 212 Marks
What is that fraction which when multiplied by itself gives 227.798649?
Answer
View full question & answer→We have to find the square root of the given number.
Hence, the fraction, which when multiplied by itself, gives 227.798649 is 15.093
Hence, the fraction, which when multiplied by itself, gives 227.798649 is 15.093
Question 222 Marks
Simplify:
$\frac{\sqrt{59.29}\ -\sqrt{5.29}}{\sqrt{59.29}\ +\sqrt{5.29}}$
$\frac{\sqrt{59.29}\ -\sqrt{5.29}}{\sqrt{59.29}\ +\sqrt{5.29}}$
Answer
View full question & answer→We have,
$\sqrt{59.29}=\sqrt{\frac{5929}{100}}=\frac{\sqrt{7\times7\times11\times11}}{10}=\frac{7\times11}{10}=7.7$
$\sqrt{59.29}=\sqrt{\frac{5929}{100}}=\frac{\sqrt{529}}{\sqrt{100}}=\frac{23}{10}=2.3$
$\frac{\sqrt{59.29}\ -\sqrt{5.29}}{\sqrt{59.29}\ +\sqrt{5.29}}=\frac{7.7-2.3}{7.7+2.3}=\frac{5.4}{10}=0.54$
$\sqrt{59.29}=\sqrt{\frac{5929}{100}}=\frac{\sqrt{7\times7\times11\times11}}{10}=\frac{7\times11}{10}=7.7$
$\sqrt{59.29}=\sqrt{\frac{5929}{100}}=\frac{\sqrt{529}}{\sqrt{100}}=\frac{23}{10}=2.3$
$\frac{\sqrt{59.29}\ -\sqrt{5.29}}{\sqrt{59.29}\ +\sqrt{5.29}}=\frac{7.7-2.3}{7.7+2.3}=\frac{5.4}{10}=0.54$
Question 232 Marks
Find the value of:
$\frac{\sqrt{1587}}{\sqrt{1728}}$
$\frac{\sqrt{1587}}{\sqrt{1728}}$
Answer
View full question & answer→We have,
$\frac{\sqrt{1587}}{\sqrt{1728}}=\sqrt{\frac{529}{576}}$ (by dividing both numbers by 3)
Computing the square roots of the numerator and the denominator,
$\sqrt{529}=\sqrt{23\times23}=23$
$\sqrt{576}=\sqrt{24\times24}=24$
$\therefore\frac{\sqrt{1587}}{\sqrt{1728}}=\frac{23}{24}$
$\frac{\sqrt{1587}}{\sqrt{1728}}=\sqrt{\frac{529}{576}}$ (by dividing both numbers by 3)
Computing the square roots of the numerator and the denominator,
$\sqrt{529}=\sqrt{23\times23}=23$
$\sqrt{576}=\sqrt{24\times24}=24$
$\therefore\frac{\sqrt{1587}}{\sqrt{1728}}=\frac{23}{24}$
Question 242 Marks
Find the square root of the following by long division method:62504836
Answer
Hence, the square root of 6250486 is 7906
View full question & answer→Hence, the square root of 6250486 is 7906Question 252 Marks
Find the square root in decimal from:
0.813604
0.813604
Answer
Hence, the square root of 0.813604 is 0.902
View full question & answer→Hence, the square root of 0.813604 is 0.902
Question 262 Marks
Find the least number which be added to the following numbers to make tham a perfect square:
$506900$
$506900$
Answer
View full question & answer→Using the long division method
,
We can see that 506900 is 44 more than $712^2$. Hence, we have to add 44 to 506900 to get a perfect square.
,
We can see that 506900 is 44 more than $712^2$. Hence, we have to add 44 to 506900 to get a perfect square.
Question 272 Marks
Find the least number which be added to the following numbers to make tham a perfect square:
5607
5607
Answer
View full question & answer→Using the long division method,
We can see that 5607 is 18 more than $75^2$. Hence, we have to add 18 to 5607 to get a perfect square.
We can see that 5607 is 18 more than $75^2$. Hence, we have to add 18 to 5607 to get a perfect square.
Question 282 Marks
Find the least number which must be subtracted from the following numbers to make tham a perfact square:
26535
26535
Answer
View full question & answer→Using the long division method,
We can see that 26535 is 291 more than $162^2$. Hence, 291 must be subtracted from 26535 to get a perfect square.
We can see that 26535 is 291 more than $162^2$. Hence, 291 must be subtracted from 26535 to get a perfect square.
Question 292 Marks
Find the squares of the following numbers using the identity $(a+b)^2=a^2+2 a b+b^2$ :
209
209
Answer
View full question & answer→$(a+b)^2=a^2+2 a b+b^2$
$(209)^2=(200+9)^2$
$=(200)^2+2 \times 200 \times 9 \times(9)^2$
$=40000+3600+81$
$=43681$
$(209)^2=(200+9)^2$
$=(200)^2+2 \times 200 \times 9 \times(9)^2$
$=40000+3600+81$
$=43681$
Question 302 Marks
Using square root table, find the square root,
11.11
11.11
Answer
View full question & answer→We have,
$\sqrt{11}=3.317$ and $\sqrt{12}=3.464$
Their difference is 0.1474
Thus, for the difference of 1 (12 - 11), the difference in the value of the square roots is 0.1474
For the difference of 0.11, the difference in the values of the square roots is,
0.11 × 0.1474 = 0.0162
$\therefore\sqrt{11.11}=3.3166+0.0162=3.328\approx3.333$
$\sqrt{11}=3.317$ and $\sqrt{12}=3.464$
Their difference is 0.1474
Thus, for the difference of 1 (12 - 11), the difference in the value of the square roots is 0.1474
For the difference of 0.11, the difference in the values of the square roots is,
0.11 × 0.1474 = 0.0162
$\therefore\sqrt{11.11}=3.3166+0.0162=3.328\approx3.333$
Question 312 Marks
Write the possible unit's digits of the square root of the following numbers. these numbers are odd square roots?
$9801$
$9801$
Answer
View full question & answer→The unit digit of the number 9801 is 1 . So, the possible unit digits are 1 or 9 (Table 3.4 ). Note that 9801 is equal to $99^2$. Hence, the square root is an odd number.
Question 322 Marks
Find the square root of the following by long division method:
1745041
1745041
Answer
Hence, the square root of 1745041 is 1321
View full question & answer→Hence, the square root of 1745041 is 1321Question 332 Marks
Using prime factorization method, find the following numbers are perfect squares?
3549
3549
Answer
View full question & answer→3549 = 3 × 7 × 13 × 13
$\begin{array}{c|c} 3& 3549 \\ \hline 7 & 1183 \\\hline 13&169 \\\hline 13&13 \\\hline &1 \end{array}$
Grouping them into pairs of equal factors,
3549 = (13 × 13) × 3 × 7
The last factors, 3 and 7 cannot be paired. Hence, 3549 is not a perfect square.
Hence, the perfect squares are 225, 441, 2916 and 11025.
$\begin{array}{c|c} 3& 3549 \\ \hline 7 & 1183 \\\hline 13&169 \\\hline 13&13 \\\hline &1 \end{array}$
Grouping them into pairs of equal factors,
3549 = (13 × 13) × 3 × 7
The last factors, 3 and 7 cannot be paired. Hence, 3549 is not a perfect square.
Hence, the perfect squares are 225, 441, 2916 and 11025.
Question 342 Marks
Find the square root in decimal from:225.6004
Answer
Hence, the square root of 225.6004 is 15.25
View full question & answer→Hence, the square root of 225.6004 is 15.25Question 352 Marks
Find the square root of the following by long division method:
12544
12544
Answer
Hence, the square root of 12544 is 112
View full question & answer→Hence, the square root of 12544 is 112
Question 362 Marks
Find the squares of the following numbers.
$425$
$425$
Answer
View full question & answer→$(425)^2$
Here $n =42$
$\therefore n(n+1)=42(42+1)$
$=42 \times 43=1806$
$\therefore(425)^2=180625$
Here $n =42$
$\therefore n(n+1)=42(42+1)$
$=42 \times 43=1806$
$\therefore(425)^2=180625$
Question 372 Marks
Find the square root in decimal form:9998.0001
Answer
Hence, the square root of 9998.001 is 99.99
View full question & answer→Hence, the square root of 9998.001 is 99.99Question 382 Marks
Observe the following pattern,
$1+3=2^2$
$1+3+5=3^2$
$1+3+5+7=4^2$
and write the value of $1+3+5+7+9+$ ...... upto n terms.
$1+3+5=3^2$
$1+3+5+7=4^2$
and write the value of $1+3+5+7+9+$ ...... upto n terms.
Answer
View full question & answer→From the pattern, we can say that the sum of the first $n$ positive odd numbers is equal to the square of the $n ^{\text {th }}$ positive number. Putting that into for $1+3+5+7+\ldots \ldots . n=n^2$, where the left hand side consists of $n$ terms.
Question 392 Marks
Find the square root in decimal form:
236.144689
236.144689
Answer
Hence, the square root of 236.144689 is 15.367
View full question & answer→Hence, the square root of 236.144689 is 15.367
Question 402 Marks
Find the square root of the following by long division method:152547201
Answer
Hence, the square root of 152547201 is 12351
View full question & answer→Hence, the square root of 152547201 is 12351Question 412 Marks
Find the least number which must be subtracted from the following numbers to make tham a perfact square:
$16160$
$16160$
Answer
View full question & answer→Using the long division method,
We can see that 16160 is 31 more than $127^2$. Hence, 31 must be subtracted from 16160 to get a perfect square.
We can see that 16160 is 31 more than $127^2$. Hence, 31 must be subtracted from 16160 to get a perfect square.
Question 422 Marks
Find the square root of the following by long division method:3915380329
Answer
Hence, the square root of 3915380329 is 625763
View full question & answer→Hence, the square root of 3915380329 is 625763Question 432 Marks
Find the squares of the following numbers.
205
205
Answer
View full question & answer→$(205)^2$
Here $n =20$
$\therefore n(n+1)=20(20+1)$
$=20 \times 21=420$
$\therefore(205)^2=42025$
Here $n =20$
$\therefore n(n+1)=20(20+1)$
$=20 \times 21=420$
$\therefore(205)^2=42025$
Question 442 Marks
Find the square root in decimal form:0.00038809
Answer
Hence, the square root of 0.00038809 is 0.0197
View full question & answer→Hence, the square root of 0.00038809 is 0.0197
Question 452 Marks
Which of the following triplets are pythagorean?
$(10,24,26)$
$(10,24,26)$
Answer
View full question & answer→A triplet ( $a, b, c$ ) is called Pythagorean if the sum of the squares of the two smallest numbers is equal to the square of the biggest number. The two smallest numbers are 10 and 24 . The sum of their squares is,
$10^2+24^2=676=26^2$
Hence, $(10,24,26)$ is a Pythagorean triplet.
$10^2+24^2=676=26^2$
Hence, $(10,24,26)$ is a Pythagorean triplet.
Question 462 Marks
Find the value of $\sqrt{103.0225}$ and hence find the value of:
$\sqrt{1.030225}$
$\sqrt{1.030225}$
Answer
View full question & answer→The value of 103.0225 is,
Hence, the square root of 103.0225 is 10.15
$\sqrt{1.030225}=\sqrt{\frac{103.0225}{100}}$
$=\frac{\sqrt{103.0225}}{\sqrt{100}}=\frac{{10.15}}{10}=1.015$
Hence, the square root of 103.0225 is 10.15
$\sqrt{1.030225}=\sqrt{\frac{103.0225}{100}}$
$=\frac{\sqrt{103.0225}}{\sqrt{100}}=\frac{{10.15}}{10}=1.015$
Question 472 Marks
Find the square root of the following by long division method:
120409
120409
Answer
Hence, the square root of 120409 is 347
View full question & answer→Hence, the square root of 120409 is 347Question 482 Marks
Using square root table, find the square root:
1312
1312
Answer
View full question & answer→Using the table to find $\sqrt{2}$ and $\sqrt{41}$$\sqrt{1312}=\sqrt{2\times2\times2\times2\times2\times41}$
$=2\times2\sqrt{2}\times\sqrt{41}$
$=2\times2\times1.414\times6.4031$
$=36.222$
$=2\times2\sqrt{2}\times\sqrt{41}$
$=2\times2\times1.414\times6.4031$
$=36.222$
Question 492 Marks
Using prime factorization method, find the following numbers are perfect squares?
343
343
Answer
View full question & answer→343 = 7 × 7 × 7
$\begin{array}{c|c} 7& 343 \\ \hline 7 & 49 \\\hline 7&7 \\\hline &1 \end{array}$
Grouping them into pairs of equal factors,
343 = (7 × 7) × 7
The last factor, 7 cannot be paired. Hence, 343 is not a perfect square.
$\begin{array}{c|c} 7& 343 \\ \hline 7 & 49 \\\hline 7&7 \\\hline &1 \end{array}$
Grouping them into pairs of equal factors,
343 = (7 × 7) × 7
The last factor, 7 cannot be paired. Hence, 343 is not a perfect square.
Question 502 Marks
Find the square root of the following by long division method:
974169
974169
Answer
Hence, the square root of 974169 is 987
View full question & answer→Hence, the square root of 974169 is 987Question 512 Marks
Find the square root of the following by long division method:
1471369
1471369
Answer
Hence, the square root of 1471369 is 1213
View full question & answer→Hence, the square root of 1471369 is 1213Question 522 Marks
Find the square root of:
$\frac{324}{841}$
$\frac{324}{841}$
Answer
View full question & answer→We know,
$\sqrt{\frac{324}{841}}=\frac{\sqrt{324}}{\sqrt{841}}$
Now, let compute the square roots of the numberator and the denominator separately.
$\sqrt{324}=\sqrt{2\times2\times3\times3\times3\times3}$
$\sqrt{841}=\sqrt{29\times29}=29$
$\therefore\sqrt{\frac{324}{841}}=\frac{81}{29}$
$\sqrt{\frac{324}{841}}=\frac{\sqrt{324}}{\sqrt{841}}$
Now, let compute the square roots of the numberator and the denominator separately.
$\sqrt{324}=\sqrt{2\times2\times3\times3\times3\times3}$
$\sqrt{841}=\sqrt{29\times29}=29$
$\therefore\sqrt{\frac{324}{841}}=\frac{81}{29}$
Question 532 Marks
Find the square root of the following by long division method:
9653449
9653449
Answer
Hence, the square root of 9653449 is 3107
View full question & answer→Hence, the square root of 9653449 is 3107Question 542 Marks
Find the squares of the following numbers using the identity $(a+b)^2=a^2+2 a b+b^2$ :
405
405
Answer
View full question & answer→$(a+b)^2=a^2+2 a b+b^2$
$(405)^2=(400+5)^2$
$=(400)^2+2 \times 400 \times 5+(5)^2$
$=160000+4000+25$
$=164025$
$(405)^2=(400+5)^2$
$=(400)^2+2 \times 400 \times 5+(5)^2$
$=160000+4000+25$
$=164025$
Question 552 Marks
Find the squares of the following numbers.
$575$
$575$
Answer
View full question & answer→$(575)^2$
$\text { Here } n=57$
$\therefore n(n+1)=57(57+1)$
$=57 \times 58=3306$
$\therefore(575)^2=330625$
$\text { Here } n=57$
$\therefore n(n+1)=57(57+1)$
$=57 \times 58=3306$
$\therefore(575)^2=330625$
Question 562 Marks
Find the square root of:
$2\frac{14}{25}$
$2\frac{14}{25}$
Answer
View full question & answer→We know,
$\sqrt{2\frac{14}{25}}=\sqrt{\frac{64}{65}}=\frac{\sqrt{64}}{\sqrt{25}}=\frac{8}{5}$
$\sqrt{2\frac{14}{25}}=\sqrt{\frac{64}{65}}=\frac{\sqrt{64}}{\sqrt{25}}=\frac{8}{5}$
Question 572 Marks
Find the value of:
$\frac{\sqrt{441}}{\sqrt{625}}$
$\frac{\sqrt{441}}{\sqrt{625}}$
Answer
View full question & answer→Computing the square roots,
$\sqrt{441}=\sqrt{(3\times3)\times(7\times7)}=3\times7=21$
$\sqrt{625}=\sqrt{(5\times5)\times(5\times5)=5\times5}=25$
$\therefore\frac{\sqrt{441}}{\sqrt{625}}=\frac{21}{25}$
$\sqrt{441}=\sqrt{(3\times3)\times(7\times7)}=3\times7=21$
$\sqrt{625}=\sqrt{(5\times5)\times(5\times5)=5\times5}=25$
$\therefore\frac{\sqrt{441}}{\sqrt{625}}=\frac{21}{25}$
Question 582 Marks
Find the square root of:
$21\frac{2797}{3364}$
$21\frac{2797}{3364}$
Answer
View full question & answer→We know, $\sqrt{21\frac{2797}{3364}}=\sqrt{\frac{73441}{3364}}=\frac{\sqrt{73441}}{\sqrt{3364}}$ Now, let us compute the square roots of the numerator and the denominator separately.
$\therefore\sqrt{21\frac{2797}{3364}}=\frac{271}{58}$
$\therefore\sqrt{21\frac{2797}{3364}}=\frac{271}{58}$
Question 592 Marks
Find the squares of the following numbers.
$405$
$405$
Answer
View full question & answer→$(405)^2$
$\text { Here } n=40$
$\therefore n(n+1)=40(40+1)$
$=40 \times 41=1640$
$\therefore(405)^2=164025$
$\text { Here } n=40$
$\therefore n(n+1)=40(40+1)$
$=40 \times 41=1640$
$\therefore(405)^2=164025$
Question 602 Marks
Find the square root of:
$25\frac{544}{729}$
$25\frac{544}{729}$
Answer
View full question & answer→We know, $\sqrt{25\frac{544}{729}}=\sqrt{\frac{18769}{729}}=\frac{\sqrt{18769}}{\sqrt{729}}$ Now, let us compute the square roots of the numerator and the denominator separately.
$\sqrt{729}=27$ $\therefore\sqrt{25\frac{544}{729}}=\frac{137}{27}$
$\sqrt{729}=27$ $\therefore\sqrt{25\frac{544}{729}}=\frac{137}{27}$
Question 612 Marks
Find the square root in decimal from:
0.7225
0.7225
Answer
Hence, the square root of 0.7225 is 0.85
View full question & answer→Hence, the square root of 0.7225 is 0.85Question 622 Marks
Find the square root of the following by long division method:
363609
363609
Answer
Hence, the square root of 363609 is 603
View full question & answer→Hence, the square root of 363609 is 603Question 632 Marks
Find the smallest number which must be added to 2300 so that it becomes a perfect square.
Answer
View full question & answer→To find the square root of 2300, we use the long division method,
23000 is $4(704-700)$ less than $48^2$. Hence, 4 must be added to 2300 to get a perfect square.
23000 is $4(704-700)$ less than $48^2$. Hence, 4 must be added to 2300 to get a perfect square.
Question 642 Marks
Which of the following triplets are pythagorean?
$(14,48,51)$
$(14,48,51)$
Answer
View full question & answer→A triplet ( $a, b, c$ ) is called Pythagorean if the sum of the squares of the two smallest numbers is equal to the square of the biggest number.
He two smallest numbers are 14 and 48 .The sum of their squares is,
$14^2+48^2=2500$, which is not equal to $51^2=2601$
Hence, $(14,48,51)$ is not a Pythagorean triplet.
He two smallest numbers are 14 and 48 .The sum of their squares is,
$14^2+48^2=2500$, which is not equal to $51^2=2601$
Hence, $(14,48,51)$ is not a Pythagorean triplet.
Question 652 Marks
Which of the following triplets are pythagorean?
$(16,63,65)$
$(16,63,65)$
Answer
View full question & answer→A triplet ( $a, b, c$ ) is called Pythagorean if the sum of the squares of the two smallest numbers is equal to the square of the biggest number. The two smallest numbers are 16 and 63 . The sum of their squares is,
$16^2+63^2=4225=65^2$
Hence, $(16,63,65)$ is a Pythagorean triplet.
$16^2+63^2=4225=65^2$
Hence, $(16,63,65)$ is a Pythagorean triplet.
Question 662 Marks
Find the squares of the following numbers using the identity $(a+b)^2=a^2+2 a b+b^2$ :
$605$
$605$
Answer
View full question & answer→$(a+b)^2=a^2+2 a b+b^2 \\
(605)^2=(600+5)^2 \\
=(600)^2+2 \times 600 \times 5 \times(5)^2 \\
=360000+6000+25 \\
=366025$
(605)^2=(600+5)^2 \\
=(600)^2+2 \times 600 \times 5 \times(5)^2 \\
=360000+6000+25 \\
=366025$
Question 672 Marks
Find the square root of:
$23\frac{394}{729}$
$23\frac{394}{729}$
Answer
View full question & answer→We know,
$\sqrt{23\frac{394}{729}}=\sqrt{\frac{17161}{729}}=\frac{\sqrt{17161}}{\sqrt{729}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$\sqrt{729}=27$
$\therefore\sqrt{23\frac{394}{729}}=\frac{131}{27}=4\frac{23}{27}$
$\sqrt{23\frac{394}{729}}=\sqrt{\frac{17161}{729}}=\frac{\sqrt{17161}}{\sqrt{729}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$\sqrt{729}=27$
$\therefore\sqrt{23\frac{394}{729}}=\frac{131}{27}=4\frac{23}{27}$
Question 682 Marks
The area of a square field is $325m^2$. Find the approximate length of one side of the field.
Answer
View full question & answer→The length of one side of the square field will be the square root of 325
$\therefore\sqrt{325}=\sqrt{5\times5\times13}$
$=5\times\sqrt{13}$
$=5\times3.605$
$=18.030$
Hence, the length of one side of the field is 18.030m
$\therefore\sqrt{325}=\sqrt{5\times5\times13}$
$=5\times\sqrt{13}$
$=5\times3.605$
$=18.030$
Hence, the length of one side of the field is 18.030m
Question 692 Marks
Find the square root in decimal form:176.252176
Answer
Hence, the square root of 0.00059049 is 0.0243
View full question & answer→Hence, the square root of 0.00059049 is 0.0243Question 702 Marks
Find the squares of the following numbers using the identity $(a-b)^2=a^2-2 a b+b^2$ :
$995$
$995$
Answer
View full question & answer→$(a-b)^2=a^2-2 a b+b^2$
$(995)^2=(1000-5)^2$
$=(1000)^2-2 \times 1000 \times 5+(5)^2$
$=1000000-10000+25$
$=1000025-10000$
$=990025$
$(995)^2=(1000-5)^2$
$=(1000)^2-2 \times 1000 \times 5+(5)^2$
$=1000000-10000+25$
$=1000025-10000$
$=990025$
Question 712 Marks
Find the squares of the following numbers.
$995$
$995$
Answer
View full question & answer→$(995)^2$
$\text { Here } n=99$
$\therefore n(n+1)=99(99+1)$
$=99 \times 100$
$=9900$
$\therefore(995)^2=990025$
$\text { Here } n=99$
$\therefore n(n+1)=99(99+1)$
$=99 \times 100$
$=9900$
$\therefore(995)^2=990025$
Question 722 Marks
Find the square root of:
$\frac{441}{961}$
$\frac{441}{961}$
Answer
View full question & answer→We know,
$\sqrt{\frac{441}{961}}=\frac{\sqrt{441}}{\sqrt{961}}$
Now, let compute the square roots of the numberator and the denominator separately.
$\sqrt{441}=\sqrt{(3\times3)\times(7\times7)}=3\times7=21$
$\sqrt{961}=\sqrt{31\times31}=31$
$\therefore\sqrt{\frac{441}{961}}=\frac{21}{31}$
$\sqrt{\frac{441}{961}}=\frac{\sqrt{441}}{\sqrt{961}}$
Now, let compute the square roots of the numberator and the denominator separately.
$\sqrt{441}=\sqrt{(3\times3)\times(7\times7)}=3\times7=21$
$\sqrt{961}=\sqrt{31\times31}=31$
$\therefore\sqrt{\frac{441}{961}}=\frac{21}{31}$
Question 732 Marks
Find the least number which be added to the following numbers to make tham a perfect square:
4931
4931
Answer
View full question & answer→Using the long division method,
We can see that 4931 is 110 more than $71^2$. Hence, we have to add 110 to 4931 to get a perfect square.
We can see that 4931 is 110 more than $71^2$. Hence, we have to add 110 to 4931 to get a perfect square.
Question 742 Marks
Using prime factorization method, find the following numbers are perfect squares?
225
225
Answer
View full question & answer→225 = 3 × 3 × 5 × 5
$\begin{array}{c|c} 3& 225 \\ \hline 3 & 75 \\\hline 5&25 \\\hline 5&5 \\\hline &1 \end{array}$
Grouping them into pairs of equal factors,
225 = (3 × 3) × (5 × 5)
There are no left out of pairs. Hence, 225 is a perfect square.
$\begin{array}{c|c} 3& 225 \\ \hline 3 & 75 \\\hline 5&25 \\\hline 5&5 \\\hline &1 \end{array}$
Grouping them into pairs of equal factors,
225 = (3 × 3) × (5 × 5)
There are no left out of pairs. Hence, 225 is a perfect square.
Question 752 Marks
Find the squares of the following numbers using the identity $(a-b)^2=a^2-2 a b+b^2$ :
99
99
Answer
View full question & answer→$(a-b)^2=a^2-2 a b+b^2$
$(99)^2=(100-1)^2$
$=(100)^2-2 \times 100 \times 1+(1)^2$
$=10000-200+1$
$=10001-200$
$=9801$
$(99)^2=(100-1)^2$
$=(100)^2-2 \times 100 \times 1+(1)^2$
$=10000-200+1$
$=10001-200$
$=9801$
Question 762 Marks
Find the value of $\sqrt{103.0225}$ and hence find the value of:
$\sqrt{10302.25}$
$\sqrt{10302.25}$
Answer
View full question & answer→The value of 103.0225 is,
Hence, the square root of 103.0225 is 10.15
$\sqrt{10302.25}=\sqrt{103.0225\times100}$
$=\sqrt{103.0225}\times{100}=10.15\times10=101.5$
Hence, the square root of 103.0225 is 10.15
$\sqrt{10302.25}=\sqrt{103.0225\times100}$
$=\sqrt{103.0225}\times{100}=10.15\times10=101.5$
Question 772 Marks
Find the square root of:
$21\frac{51}{169}$
$21\frac{51}{169}$
Answer
View full question & answer→We know,
$\sqrt{21\frac{51}{169}}=\sqrt{\frac{3600}{169}}=\frac{\sqrt{3600}}{\sqrt{169}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$\sqrt{3600}=\sqrt{60\times60}=60$
$\sqrt{169}=\sqrt{13\times13}=13$
$\therefore\sqrt{21\frac{51}{169}}=\frac{60}{13}=4\frac{8}{13}$
$\sqrt{21\frac{51}{169}}=\sqrt{\frac{3600}{169}}=\frac{\sqrt{3600}}{\sqrt{169}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$\sqrt{3600}=\sqrt{60\times60}=60$
$\sqrt{169}=\sqrt{13\times13}=13$
$\therefore\sqrt{21\frac{51}{169}}=\frac{60}{13}=4\frac{8}{13}$
Question 782 Marks
Find the square root of the following by long division method:6407522209
Answer
Hence, the square root of 6407522209 is 80047
View full question & answer→Hence, the square root of 6407522209 is 80047Question 792 Marks
Find the least number of three digits which is perfect square.
Answer
View full question & answer→Let us make a list of the squares starting from 1.
$1^2=1$
$2^2=4$
$3^2=9$
$4^2=16$
$5^2=25$
$6^2=36$
$7^2=49$
$8^2=64$
$9^2=81$
$10^2=100$
The square of 10 has three digits. Hence, the least three-digit perfect square is 100
$1^2=1$
$2^2=4$
$3^2=9$
$4^2=16$
$5^2=25$
$6^2=36$
$7^2=49$
$8^2=64$
$9^2=81$
$10^2=100$
The square of 10 has three digits. Hence, the least three-digit perfect square is 100
Question 802 Marks
Using prime factorization method, find the following numbers are perfect squares?
11025
11025
Answer
View full question & answer→11025 = 3 × 3 × 5 × 5 × 7 × 7
$\begin{array}{c|c} 3& 11025 \\ \hline 3 & 3675 \\\hline 5&1225 \\\hline 5&245 \\\hline 7&49 \\\hline 7&7 \\\hline &1 \end{array}$
Grouping them into pairs of equal factors,
11025 = (3 × 2) × (5 × 5) × (7 × 7)
There are no left out of pairs. Hence, 11025 is a perfect square.
$\begin{array}{c|c} 3& 11025 \\ \hline 3 & 3675 \\\hline 5&1225 \\\hline 5&245 \\\hline 7&49 \\\hline 7&7 \\\hline &1 \end{array}$
Grouping them into pairs of equal factors,
11025 = (3 × 2) × (5 × 5) × (7 × 7)
There are no left out of pairs. Hence, 11025 is a perfect square.
Question 812 Marks
Find the least number which be added to the following numbers to make tham a perfect square:
37460
37460
Answer
View full question & answer→Using the long division method,
We can see that 194491 is 10 more than $441^2$. Hence, 10 must be subtracted from 194491 to get a perfact square.
We can see that 194491 is 10 more than $441^2$. Hence, 10 must be subtracted from 194491 to get a perfact square.
Question 822 Marks
What is the fraction which when multiplied by itself gives 0.00053361?
Answer
View full question & answer→We have to find the square root of the given number.
Hence, the fraction which multiplied by itself, gives 0.00053361 is 0.0231
Hence, the fraction which multiplied by itself, gives 0.00053361 is 0.0231
Question 832 Marks
Find the squares of the following numbers:
$127$
$127$
Answer
View full question & answer→$(127)^2=(120+7)^2$
$\left\{(a+b)^2=a^2+2 a b+b^2\right\}$
$=(120)^2+2 \times 120 \times 7+(7)^2$
$=14400+1680+49$
$=16129$
$\left\{(a+b)^2=a^2+2 a b+b^2\right\}$
$=(120)^2+2 \times 120 \times 7+(7)^2$
$=14400+1680+49$
$=16129$
Question 842 Marks
Find the least number which must be subtracted from the following numbers to make tham a perfact square:
$2361$
$2361$
Answer
View full question & answer→Using the long division method,
We can see that 2361 is 57 more than $47^2$. Hence, 57 must be subtracted from 2361 to get perfact square.
We can see that 2361 is 57 more than $47^2$. Hence, 57 must be subtracted from 2361 to get perfact square.
Question 852 Marks
Find the square root of the following by long division method:20421361
Answer
Hence, the square root of 20421361 is 4519
View full question & answer→Hence, the square root of 20421361 is 4519Question 862 Marks
Find the squares of the following numbers using the identity $(a-b)^2=a^2-2 a b+b^2$ :
$395$
$395$
Answer
View full question & answer→$(a-b)^2=a^2-2 a b+b^2$
$(395)^2=(400-5)^2$
$=(400)^2-2 \times 400 \times 5+(5)^2$
$=160000-4000+25$
$=160025-4000$
$=156025$
$(395)^2=(400-5)^2$
$=(400)^2-2 \times 400 \times 5+(5)^2$
$=160000-4000+25$
$=160025-4000$
$=156025$
Question 872 Marks
Find the square root of the following by long division method:20657025
Answer
Hence, the square root of 20657025 is 4545
View full question & answer→Hence, the square root of 20657025 is 4545Question 882 Marks
Find the square root of the following by long division method:
390625
390625
Answer
Hence, the square root of 390625 is 625
View full question & answer→Hence, the square root of 390625 is 625Question 892 Marks
Find the squares of the following numbers:
$265$
$265$
Answer
View full question & answer→$(265)^2=(200+65)^2$
$\left\{(a+b)^2=a^2+2 a b+b^2\right\}$
$=(200)^2+2 \times 200 \times 65+(65)^2$
$=40000+26000+4225$
$=70225$
$\left\{(a+b)^2=a^2+2 a b+b^2\right\}$
$=(200)^2+2 \times 200 \times 65+(65)^2$
$=40000+26000+4225$
$=70225$
Question 902 Marks
Find the square root in decimal from:
84.8241
84.8241
Answer
Hence, the square root of 84.8241 is 9.21
View full question & answer→Hence, the square root of 84.8241 is 9.21
Question 912 Marks
Find the square root in decimal from:150.0625
Answer
Hence, the square root of 150.0625 is 12.25
View full question & answer→Hence, the square root of 150.0625 is 12.25Question 922 Marks
Find the square root of:
$23\frac{26}{121}$
$23\frac{26}{121}$
Answer
View full question & answer→We know,
$\sqrt{23\frac{26}{121}}=\sqrt{\frac{2809}{121}}=\frac{\sqrt{2809}}{\sqrt{121}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$\sqrt{121}=11$
$\therefore\sqrt{23\frac{26}{121}}=\frac{53}{11}$
$\sqrt{23\frac{26}{121}}=\sqrt{\frac{2809}{121}}=\frac{\sqrt{2809}}{\sqrt{121}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$\sqrt{121}=11$
$\therefore\sqrt{23\frac{26}{121}}=\frac{53}{11}$
Question 932 Marks
Find the square root of the following by long division method:
286225
286225
Answer
Hence, the square root of 286225 is 535
View full question & answer→Hence, the square root of 286225 is 535Question 942 Marks
Find the greatest number of two digits which is a perfect square.
Answer
View full question & answer→We know that $10^2$ is equal to 100 and $9^2$ is equal to 81 . Since 10 and 9 are consecutive numbers, there is no perfect square between 100 and 81 . Since 100 is the first perfect square that has more than two digits, 81 is the greatest two-digit perfect square.
Question 952 Marks
Find the square root of the following by long division method:
97344
97344
Answer
Hence, the square root of 97344 is 312
View full question & answer→Hence, the square root of 97344 is 312Question 962 Marks
Write the possible unit's digits of the square root of the following numbers. these numbers are odd square roots?
657666025
657666025
Answer
View full question & answer→The unit digit of the number 657666025 is 5 . So, the only possible unit digit is 5 . Note that 657666025 is equal to $(5 \times 23 \times 223)^2$. Hence, the square root is an odd number.
Question 972 Marks
Find the square root of:
$4\frac{29}{49}$
$4\frac{29}{49}$
Answer
View full question & answer→We know,
$\sqrt{4\frac{29}{49}}=\sqrt{\frac{225}{49}}=\frac{\sqrt{225}}{\sqrt{49}}$
$\sqrt{225}=15$
$\sqrt{49}=7$
$\therefore\sqrt{4\frac{29}{49}}=\frac{15}{7}$
$\sqrt{4\frac{29}{49}}=\sqrt{\frac{225}{49}}=\frac{\sqrt{225}}{\sqrt{49}}$
$\sqrt{225}=15$
$\sqrt{49}=7$
$\therefore\sqrt{4\frac{29}{49}}=\frac{15}{7}$
Question 982 Marks
Write the possible unit's digits of the square root of the following numbers. these numbers are odd square roots?
99856
99856
Answer
View full question & answer→The unit digit of the number 99856 is 6. So, the possible unit digits are 4 or 6 (Table 3.4). Since its last digit is 6 (an even number), it cannot have an odd number as its square root.
Question 992 Marks
Find the square root of:
$3\frac{334}{3025}$
$3\frac{334}{3025}$
Answer
View full question & answer→We know,
$\sqrt{3\frac{334}{3025}}=\sqrt{\frac{9409}{3025}}=\frac{\sqrt{9409}}{\sqrt{3025}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$\therefore\sqrt{3\frac{334}{3035}}=\frac{97}{55}$
$\sqrt{3\frac{334}{3025}}=\sqrt{\frac{9409}{3025}}=\frac{\sqrt{9409}}{\sqrt{3025}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$\therefore\sqrt{3\frac{334}{3035}}=\frac{97}{55}$
Question 1002 Marks
Find the squares of the following numbers using the identity $(a-b)^2=a^2-2 a b+b^2$ :
$495$
$495$
Answer
View full question & answer→$(a-b)^2=a^2-2 a b+b^2$
$(495)^2=(500-5)^2$
$=(500)^2-2 \times 500 \times 5+(5)^2$
$=250000-5000+25$
$=250025-5000$
$=245025$
$(495)^2=(500-5)^2$
$=(500)^2-2 \times 500 \times 5+(5)^2$
$=250000-5000+25$
$=250025-5000$
$=245025$
Question 1012 Marks
Find the square root of the following by long division method:3226694416
Answer
Hence, the square root of 3226694416 is 56804
View full question & answer→Hence, the square root of 3226694416 is 56804Question 1022 Marks
Find the squares of the following numbers.
745
745
Answer
View full question & answer→$(745)^2$
Here $n =74$
$\therefore n(n+1)=74(74+1)$
$=74 \times 75=5550$
$\therefore(745)^2=555025$
Here $n =74$
$\therefore n(n+1)=74(74+1)$
$=74 \times 75=5550$
$\therefore(745)^2=555025$
Question 1032 Marks
Using prime factorization method, find the following numbers are perfect squares?
189
189
Answer
View full question & answer→189 = 3 × 3 × 3 × 7
$\begin{array}{c|c} 3& 189 \\ \hline 3 & 63 \\\hline 3&21 \\\hline 7&7 \\\hline &1 \end{array}$
Grouping them into pairs of equal factors,
189 = (3 × 3) × 3 × 7
The factors 3 and 7 cannot be paired. Hence, 189 is not a perfect square.
$\begin{array}{c|c} 3& 189 \\ \hline 3 & 63 \\\hline 3&21 \\\hline 7&7 \\\hline &1 \end{array}$
Grouping them into pairs of equal factors,
189 = (3 × 3) × 3 × 7
The factors 3 and 7 cannot be paired. Hence, 189 is not a perfect square.
Question 1042 Marks
Find the square root of the following correct to three places of decimal:
$2\frac{1}{2}$
$2\frac{1}{2}$
Answer
View full question & answer→We can find the square root up to four decimal places by expanding $2\frac{1}{2}$ into decimal form up to eight digits to the right of the decimal point as shown below,
$2\frac{1}{2}=2.50000000$
But, this is the same with the value 2.5 in problem (ix). Hence, the square root of $2\frac{1}{2}$ is 1.581
$2\frac{1}{2}=2.50000000$
But, this is the same with the value 2.5 in problem (ix). Hence, the square root of $2\frac{1}{2}$ is 1.581
Question 1052 Marks
Find the square root of:
$38\frac{11}{25}$
$38\frac{11}{25}$
Answer
View full question & answer→We know,
$\sqrt{38\frac{11}{25}}=\sqrt{\frac{961}{25}}=\frac{\sqrt{961}}{\sqrt{25}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$\sqrt{961}=31$
$\sqrt{25}=5$
$\therefore\sqrt{38\frac{11}{25}}=\frac{31}{5}$
$\sqrt{38\frac{11}{25}}=\sqrt{\frac{961}{25}}=\frac{\sqrt{961}}{\sqrt{25}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$\sqrt{961}=31$
$\sqrt{25}=5$
$\therefore\sqrt{38\frac{11}{25}}=\frac{31}{5}$
Question 1062 Marks
Find the squares of the following numbers:
$862$
$862$
Answer
View full question & answer→$(862)^2=(800+62)^2$
$\left\{(a+b)^2=a^2+2 a b+b^2\right\}$
$=(800)^2+2 \times 800 \times 62+(62)^2$
$=640000+99200+3844$
$=743044$
$\left\{(a+b)^2=a^2+2 a b+b^2\right\}$
$=(800)^2+2 \times 800 \times 62+(62)^2$
$=640000+99200+3844$
$=743044$
Question 1072 Marks
Find the square root in decimal form:0.00059049
Answer
Hence, the square root of 0.00059049 is 0.0243
View full question & answer→Hence, the square root of 0.00059049 is 0.0243
Question 1082 Marks
Find the least number which must be subtracted from the following numbers to make tham a perfact square:
$4401624$
$4401624$
Answer
View full question & answer→Using the long division method,
We can see that 4401624 is 20 more than $2098^2$. Hence, 20 must be subtracted from 4401624 to get a perfect square.
We can see that 4401624 is 20 more than $2098^2$. Hence, 20 must be subtracted from 4401624 to get a perfect square.
Question 1092 Marks
Find the square root of the following by long division method:
291600
291600
Answer
Hence, the square root of 291600 is 540
View full question & answer→Hence, the square root of 291600 is 540Question 1102 Marks
Find the squares of the following numbers using the identity $(a+b)^2=a^2+2 a b+b^2$ :
1001
1001
Answer
View full question & answer→$(a+b)^2=a^2+2 a b+b^2$
$(1001)^2=(1000+1)^2$
$=(1000)^2+2 \times 1000 \times 1 \times(1)$
$=1000000+2000+1$
$=1002001$
$(1001)^2=(1000+1)^2$
$=(1000)^2+2 \times 1000 \times 1 \times(1)$
$=1000000+2000+1$
$=1002001$
Question 1112 Marks
Find the squares of the following numbers using the identity $(a-b)^2=a^2-2 a b+b^2$ :
$599$
$599$
Answer
View full question & answer→$(a-b)^2=a^2-2 a b+b^2$
$(599)^2=(600-1)^2$
$=(600)^2-2 \times 600 \times 1+(1)^2$
$=360000-1200+1$
$=360001-1200$
$=358801$
$(599)^2=(600-1)^2$
$=(600)^2-2 \times 600 \times 1+(1)^2$
$=360000-1200+1$
$=360001-1200$
$=358801$
Question 1122 Marks
Find the squares of the following numbers using the identity $(a-b)^2=a^2-2 a b+b^2$ :
$999$
$999$
Answer
View full question & answer→$(a-b)^2=a^2-2 a b+b^2$
$(999)^2=(1000-1)^2$
$=(1000)^2-2 \times 1000 \times 1+(1)^2$
$=1000000-2000+1$
$=10000001-2000$
$=998001$
$(999)^2=(1000-1)^2$
$=(1000)^2-2 \times 1000 \times 1+(1)^2$
$=1000000-2000+1$
$=10000001-2000$
$=998001$
Question 1132 Marks
Find the least number which must be subtracted from the following numbers to make tham a perfact square:
$194491$
$194491$
Answer
View full question & answer→Using the long division method,
We can see that 194491 is 10 more than $441^2$. Hence, 10 must be subtracted from 194491 to get a perfact square.
We can see that 194491 is 10 more than $441^2$. Hence, 10 must be subtracted from 194491 to get a perfact square.
Question 1142 Marks
Write the possible unit's digits of the square root of the following numbers. these numbers are odd square roots?
998001
998001
Answer
View full question & answer→The unit digit of the number 998001 is 1 . So, the possible unit digits are 1 or 9 . Note that 998001 is equal to $\left(3^3 \times 37\right)^2$. Hence, the square root is an odd number.
Question 1152 Marks
Which of the following triplets are pythagorean?
$(14,35,38)$
$(14,35,38)$
Answer
View full question & answer→A triplet ( $a, b, c$ ) is called Pythagorean if the sum of the squares of the two smallest numbers is equal to the square of the biggest number.
The two smallest numbers are 12 and 35 . The sum of their squares is,
$12^2+35^2=1369$, which is not equal to $38^2=1444$
Hence, $(12,35,38)$ is not a Pythagorean triplet.
The two smallest numbers are 12 and 35 . The sum of their squares is,
$12^2+35^2=1369$, which is not equal to $38^2=1444$
Hence, $(12,35,38)$ is not a Pythagorean triplet.
Question 1162 Marks
Find the squares of the following numbers:
$503$
$503$
Answer
View full question & answer→$(503)^2=(500+3)^2$
$\left\{(a+b)^2=a^2+2 a b+b^2\right\}$
$=(500)^2+2 \times 500 \times 3+(3)^2$
$=250000+3000+9$
$=253009$
$\left\{(a+b)^2=a^2+2 a b+b^2\right\}$
$=(500)^2+2 \times 500 \times 3+(3)^2$
$=250000+3000+9$
$=253009$
Question 1172 Marks
Using prime factorization method, find the following numbers are perfect squares?
2048
2048
Answer
View full question & answer→2048 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
$\begin{array}{c|c} 2& 2048 \\ \hline 2 & 1024 \\\hline 2&512 \\\hline 2&256 \\\hline 2&128 \\\hline 2&64 \\\hline 2&32 \\\hline2&16 \\\hline2&8 \\\hline2&4 \\\hline2&2 \\\hline&1 \end{array}$
Grouping them into pairs of equal factors,
2048 = (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × 2
The last factor, 2 cannot be paired. Hence, 2048 is not a perfect square.
$\begin{array}{c|c} 2& 2048 \\ \hline 2 & 1024 \\\hline 2&512 \\\hline 2&256 \\\hline 2&128 \\\hline 2&64 \\\hline 2&32 \\\hline2&16 \\\hline2&8 \\\hline2&4 \\\hline2&2 \\\hline&1 \end{array}$
Grouping them into pairs of equal factors,
2048 = (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × 2
The last factor, 2 cannot be paired. Hence, 2048 is not a perfect square.
Question 1182 Marks
Using prime factorization method, find the following numbers are perfect squares?
2916
2916
Answer
View full question & answer→2916 = 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3
$\begin{array}{c|c} 2& 2916 \\ \hline 2 & 1458 \\\hline 3&729 \\\hline 3&243 \\\hline 3&81 \\\hline 3&27 \\\hline 3&9 \\\hline 3&3 \\\hline &1 \end{array}$
Grouping them into pairs of equal factors,
2916 = (2 × 2) × (3 × 3) × (3 × 3) × (3 × 3)
There are no left out of pairs. Hence, 2916 is a perfect square.
$\begin{array}{c|c} 2& 2916 \\ \hline 2 & 1458 \\\hline 3&729 \\\hline 3&243 \\\hline 3&81 \\\hline 3&27 \\\hline 3&9 \\\hline 3&3 \\\hline &1 \end{array}$
Grouping them into pairs of equal factors,
2916 = (2 × 2) × (3 × 3) × (3 × 3) × (3 × 3)
There are no left out of pairs. Hence, 2916 is a perfect square.