Question 512 Marks
Find the square root of the following by long division method: $1471369$
Answer
Hence, the square root of $1471369$ is $1213$
View full question & answer→Hence, the square root of $1471369$ is $1213$
Questions · Page 2 of 3
2 Marks Questions
Question 522 Marks
Find the square root of: $\frac{324}{841}$
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. $\sqrt{324}=\sqrt{2\times2\times3\times3\times3\times3}$
$\sqrt{841}=\sqrt{29\times29}=29$
$\therefore\sqrt{\frac{324}{841}}=\frac{81}{29}$
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: $9653449$
Answer
Hence, the square root of $9653449$ is $3107$
View full question & answer→Hence, the square root of $9653449$ is $3107$
Question 542 Marks
Find the squares of the following numbers using the identity $ (a+b)^2=a^2+2 a b+b^2 : 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. $575$
Answer
View full question & answer→$(575)^2$
Here $n = 57$
$\therefore$ $n(n + 1) = 57(57 + 1)$
$= 57 \times 58 = 3306$
$\therefore$ $(575)^2$$ = 330625$
Here $n = 57$
$\therefore$ $n(n + 1) = 57(57 + 1)$
$= 57 \times 58 = 3306$
$\therefore$ $(575)^2$$ = 330625$
Question 562 Marks
Find the square root of: $2\frac{14}{25}$
Answer
View full question & answer→We know, $\sqrt{2\frac{14}{25}}=\sqrt{\frac{64}{65}}=\frac{\sqrt{64}}{\sqrt{25}}=\frac{8}{5}$
Question 572 Marks
Find the value of: $\frac{\sqrt{441}}{\sqrt{625}}$
Answer
View full question & answer→Computing the square roots, $\sqrt{441}=\sqrt{(3\times3)\times(7\times7)}=3\times7=21$ $\sqrt{625}=\sqrt{(5\times5)\times(5\times5)=5\times5}=25$
$\therefore\frac{\sqrt{441}}{\sqrt{625}}=\frac{21}{25}$
$\therefore\frac{\sqrt{441}}{\sqrt{625}}=\frac{21}{25}$
Question 582 Marks
Find the square root of: $21\frac{2797}{3364}$
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.
$\therefore\sqrt{21\frac{2797}{3364}}=\frac{271}{58}$
Now, let us compute the square roots of the numerator and the denominator separately.
$\therefore\sqrt{21\frac{2797}{3364}}=\frac{271}{58}$
Question 592 Marks
Find the squares of the following numbers. $405$
Answer
View full question & answer→$(405)^2$
Here $n = 40$
$\therefore$ $n(n + 1) = 40(40 + 1)$
$= 40 \times 41 = 1640$
$\therefore$ $(405)^2= 164025$
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: $25\frac{544}{729}$
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.
$\sqrt{729}=27$ $\therefore\sqrt{25\frac{544}{729}}=\frac{137}{27}$
Now, let us compute the square roots of the numerator and the denominator separately.
$\sqrt{729}=27$ $\therefore\sqrt{25\frac{544}{729}}=\frac{137}{27}$
Question 612 Marks
Find the square root in decimal from: $0.7225$
Answer
Hence, the square root of $0.7225$ is $0.85$
View full question & answer→Hence, the square root of $0.7225$ is $0.85$
Question 622 Marks
Find the square root of the following by long division method: $363609$
Answer
Hence, the square root of $363609$ is $603$
View full question & answer→Hence, the square root of $363609$ is $603$
Question 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? $(14, 48, 51)$
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? $(16, 63, 65)$
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.
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.
Question 662 Marks
Find the squares of the following numbers using the identity $(a + b)^2= a^2+ 2ab + b^2: 605$
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: $23\frac{394}{729}$
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.
$\sqrt{729}=27$
$\therefore\sqrt{23\frac{394}{729}}=\frac{131}{27}=4\frac{23}{27}$
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 $18.030\ m$
$\therefore\sqrt{325}=\sqrt{5\times5\times13}$
$=5\times\sqrt{13}$
$=5\times3.605$
$=18.030$
Hence, the length of one side of the field is $18.030\ m$
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 $0.0243$.
Question 702 Marks
Find the squares of the following numbers using the identity $ (a-b)^2=a^2-2 a b+b^2 : 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. $995$
Answer
View full question & answer→$(995)^2$
Here $n = 99$
$\therefore$ $n(n + 1) = 99(99 + 1)$
$= 99 × 100 = 9900$
$\therefore$ $(995)^2= 990025$
Here $n = 99$
$\therefore$ $n(n + 1) = 99(99 + 1)$
$= 99 × 100 = 9900$
$\therefore$ $(995)^2= 990025$
Question 722 Marks
Find the square root of: $\frac{441}{961}$
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. $\sqrt{441}=\sqrt{(3\times3)\times(7\times7)}=3\times7=21$
$\sqrt{961}=\sqrt{31\times31}=31$
$\therefore\sqrt{\frac{441}{961}}=\frac{21}{31}$
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: $4931$
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? $225$
Answer
View full question & answer→$225 = 3 \times 3 \times 5 \times 5$
$\begin{array}{c|c} 3& 225 \\ \hline 3 & 75 \\\hline 5&25 \\\hline 5&5 \\\hline &1 \end{array}$
Grouping them into pairs of equal factors, $225 = (3 \times 3) \times (5 \times 5)$
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, $225 = (3 \times 3) \times (5 \times 5)$
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 $(a-b)^2=a^2-2 a b+b^2: 99$
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: $\sqrt{10302.25}$
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: $21\frac{51}{169}$
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.
$\sqrt{3600}=\sqrt{60\times60}=60$
$\sqrt{169}=\sqrt{13\times13}=13$
$\therefore\sqrt{21\frac{51}{169}}=\frac{60}{13}=4\frac{8}{13}$
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 $80047$
Question 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? $11025$
Answer
View full question & answer→$11025 = 3 \times 3 \times 5 \times 5 \times 7 \times 7$
$\begin{array}{c|c} 3& 11025 \\ \hline 3 & 3675 \\\hline 5&1225 \\\hline 5&245 \\\hline 7&49 \\\hline 7&7 \\\hline &1 \end{array}$
Grouping them into pairs of equal factors,
$11025 = (3 \times 2) \times (5 \times 5) \times (7 \times 7)$
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,
$11025 = (3 \times 2) \times (5 \times 5) \times (7 \times 7)$
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: $37460$
Answer
View full question & answer→Using the long division method,
We can see that $37460$ is $176$ more than $194^2$.
Hence, we have to add $176$ to $37460$ to get a perfect square.
We can see that $37460$ is $176$ more than $194^2$.
Hence, we have to add $176$ to $37460$ to get a perfect 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: $127$
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: $2361$
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 $4519$.
Question 862 Marks
Find the squares of the following numbers using the identity $(a-b)^2=a^2-2 a b+b^2 : 395$
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 $4545$
Question 882 Marks
Find the square root of the following by long division method: $390625$
Answer
Hence, the square root of $390625$ is $625$.
View full question & answer→Hence, the square root of $390625$ is $625$.
Question 892 Marks
Find the squares of the following numbers: $265$
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: $84.8241$
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 $12.25$.
Question 922 Marks
Find the square root of: $23\frac{26}{121}$
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.
$\sqrt{121}=11$
$\therefore\sqrt{23\frac{26}{121}}=\frac{53}{11}$
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 $535$
Question 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.
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: $97344$
Answer
Hence, the square root of $97344$ is $312$.
View full question & answer→Hence, the square root of $97344$ is $312$.
Question 962 Marks
Write the possible unit's digits of the square root of the following numbers. these numbers are odd square roots? $657666025$
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.
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: $4\frac{29}{49}$
Answer
View full question & answer→We know, $\sqrt{4\frac{29}{49}}=\sqrt{\frac{225}{49}}=\frac{\sqrt{225}}{\sqrt{49}}$
$\sqrt{225}=15$ $\sqrt{49}=7$
$\therefore\sqrt{4\frac{29}{49}}=\frac{15}{7}$
$\sqrt{225}=15$ $\sqrt{49}=7$
$\therefore\sqrt{4\frac{29}{49}}=\frac{15}{7}$
Question 982 Marks
Write the possible unit's digits of the square root of the following numbers. these numbers are odd square roots? $99856$
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.
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: $3\frac{334}{3025}$
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.
$\therefore\sqrt{3\frac{334}{3035}}=\frac{97}{55}$
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 $
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 $