Question 11 Mark
Without actual division, find the following rational numbers are terminating decimals. $\frac{16}{125}$
Answer$\frac{16}{125}$ Denominator of $\frac{16}{125}$ is $125$ . And, $125=5^3$ Therefore, $125$ has no other factors than $2$ and $5$ . Thus, $\frac{16}{125}$ is a terminating decimal.
View full question & answer→Question 21 Mark
What can you say about the sum of a rational number and an irrational number?
AnswerThe sum of a rational number and an irrational number is irrational. Example: $5+\sqrt{3}$ is irrational.
View full question & answer→Question 31 Mark
The number $\frac{665}{625}$ will terminate after how many decimal places?
Answer
Thus, the given number will terminate after $3$ decimal places. View full question & answer→Question 41 Mark
Classify the following number as rational or irrational. give reasons to support your answer. $4.1276$
Answer$4.1276$ It is a terminating decimal. Hence, it is rational.
View full question & answer→Question 51 Mark
Classify the following number as rational or irrational. give reasons to support your answer. $6.834837...$
Answer$6.834837...$ It is neither terminating, nor repeating hence it is irrational number.
View full question & answer→Question 61 Mark
Without actual division, find the following rational numbers are terminating decimals.
$\frac{31}{375}$
Answer$\frac{31}{375}$
Denominator of $\frac{31}{375}$ is $375 .$
$375=5^3 \times 3$
So, the prime factor $375$ are $5$ and $3 .$
Thus, $\frac{31}{375}$ is not a terminating decimal.
View full question & answer→Question 71 Mark
Evaluate:
$\big(125\big)^{\frac{1}{3}}$
Answer $\big(125\big)^{\frac{1}{3}}=(5^3)^{\frac{1}{3}}=5^{3\times\frac{1}{3}}=5^1=5$
View full question & answer→Question 81 Mark
Classify the following number as rational or irrational. give reasons to support your answer. $3.040040004...$
Answer$3.040040004...$ is an irrational number because it is a non-terminating, non-repeating decimal.
View full question & answer→Question 91 Mark
Rationalise the denominator of the following: $\frac{1}{\sqrt{7}}$
AnswerOn multiplying the numerator and denominator of the given number by $\sqrt{7},$ we get $\frac{1}{\sqrt{7}}=\frac{1}{\sqrt{7}}\times\frac{\sqrt{7}}{\sqrt{7}}=\frac{\sqrt{7}}{7}.$
View full question & answer→Question 101 Mark
Without actual division, find the following rational numbers are terminating decimals.
$\frac{5}{12}$
Answer$\frac{5}{12}$
Denominator of $\frac{5}{12}$ is $12 .$
And,
$12=2^2 \times 3$
So, $12$ has a prime factor $3$ , which is other than $2$ and $5 .$
Thus, $\frac{5}{12}$ is not a terminating decimal.
View full question & answer→Question 111 Mark
Rationalise $\frac{1}{\sqrt{3}+\sqrt{2}}.$
Answer $\frac{1}{\sqrt{3}+\sqrt{2}}=\frac{1}{\sqrt{3}+\sqrt{2}}\times\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}$
$=\frac{\sqrt{3}-\sqrt{2}}{\big(\sqrt{3}\big)^2-\big(\sqrt{2}\big)^2}$
$=\frac{\sqrt{3}-\sqrt{2}}{3-2}$
$=\sqrt{3}-\sqrt{2}$
View full question & answer→Question 121 Mark
Represent the following rational numbers on the number line: $-2.4$
Answer$-2.4=\frac{-24}{10}=\frac{-12}{5}=-2\frac{2}{5}$
View full question & answer→Question 131 Mark
Find two irrational numbers between $0.16$ and $0.17.$
AnswerTwo irrational numbers between $0.16$ and $0.17$ are as follows: $0.1611161111611111611111...$ and $0.169669666...$
View full question & answer→Question 141 Mark
Evaluate: $\big(64\big)^{\frac{1}{6}}$
Answer$\big(64\big)^{\frac{1}{6}}=(2^6)^{\frac{1}{6}}=2^{\big(6\times\frac{1}{6}\big)}=2^1=2$
View full question & answer→Question 151 Mark
Solve: $\big(3-\sqrt{11}\big)\big(3+\sqrt{11}\big).$
Answer$\big(3-\sqrt{11}\big)\big(3+\sqrt{11}\big)$ $=3^2-\big(\sqrt{11}\big)^2$ $=9-11$ $=-2$
View full question & answer→Question 161 Mark
Without actual division, find the following rational numbers are terminating decimals.
$\frac{7}{24}$
Answer$\frac{7}{24}$
Denominator of $\frac{7}{24}$ is $24 .$
And,
$24=2^3 \times 3$
So, $24$ has a prime factor $3 $, which is other than $2$ and $5 .$
Thus, $\frac{7}{24}$ is not a terminating decimal.
View full question & answer→Question 171 Mark
Simplify: $\frac{6^{\frac{1}{4}}}{6^{\frac{1}{5}}}$
Answer$\frac{6^{\frac{1}{4}}}{6^{\frac{1}{5}}}=6^{\big(\frac{1}{4}-\frac{1}{5}\big)}$ $=6^{\big(\frac{5-4}{20}\big)}=6^{\frac{1}{20}}$
View full question & answer→Question 181 Mark
Classify the following number as rational or irrational. give reasons to support your answer.
$\frac{22}{7}$
Answer $\frac{22}{7}$ is a rational number because it can be expressed in the $\frac{\text{p}}{\text{q}}$ form.
View full question & answer→Question 191 Mark
Classify the following number as rational or irrational. give reasons to support your answer. $\sqrt{1.44}$
Answer$\sqrt{1.44}=1.2$ So, it is rational.
View full question & answer→Question 201 Mark
Simplify: $(14641)^{0.25}$
Answer$(14641)^{0.25}$ $=(14641)^{\frac{1}{4}}$ $=(11^4)^{\frac{1}{4}}$ $=11^{4\times\frac{1}{4}}$ $=11$
View full question & answer→Question 211 Mark
Evaluate: $\big(64\big)^{-\frac{1}{2}}$
Answer$\big(64\big)^{-\frac{1}{2}}=\frac{1}{\big(64\big)^{\frac{1}{2}}}=\frac{1}{\big(8^2\big)^{\frac{1}{2}}}=\frac{1}{\big(8\big)^{2\times\frac{1}{2}}}$ $=\frac{1}{8^1}=\frac{1}{8}$
View full question & answer→Question 221 Mark
Classify the following number as rational or irrational. give reasons to support your answer.
$\frac{2}{3}\sqrt{6}$
Answer $\frac{2}{3}\sqrt{6}$
It is an irrational number.
View full question & answer→Question 231 Mark
Give an example of two irrational numbers whose: Quotient is an irrational number.
Answer$2$ irrational numbers with quotient an irrational number will be $\sqrt{15}$ and $\sqrt{5}$
View full question & answer→Question 241 Mark
Find an irrational number between $5$ and $6.$
AnswerAn irrational number between $5$ and $6 =\sqrt{5\times6}=\sqrt{30}$
View full question & answer→Question 251 Mark
Give an example of two irrational numbers whose: Product is an irrational number.
Answer$2$ irrational numbers with product an irrational number will be $6+\sqrt{3}$ and $7-\sqrt{3}$
View full question & answer→Question 261 Mark
Give an example of two irrational numbers whose: Sum is an irrational number.
Answer$2$ irrational numbers with sum an irrational number $7+\sqrt{5}$ and $\sqrt{6}-8$
View full question & answer→Question 271 Mark
Rationalise the denominator of the following:
$\frac{\sqrt{5}}{2\sqrt{3}}$
Answer On multiplying the numerator and denominator of the given number by $\sqrt{3},$ we get
$\frac{\sqrt{5}}{2\sqrt{3}}=\frac{\sqrt{5}}{2\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{15}}{2\times3}=\frac{\sqrt{15}}{6}$
View full question & answer→Question 281 Mark
Simplify: $6^\frac{1}{2}\times7^\frac{1}{2}$
Answer$6^\frac{1}{2}\times7^\frac{1}{2}=(6\times7)^{\frac{1}{2}}=(42)^{\frac{1}{2}}$
View full question & answer→Question 291 Mark
Represent the following rational numbers on the number line: $\frac{5}{7}$
Answer$\frac{5}{7}$ 
View full question & answer→Question 301 Mark
Simplify $\Big(\frac{3125}{243}\Big)^{\frac{4}{5}}.$
Answer$\Big(\frac{3125}{243}\Big)^{\frac{4}{5}}$ $=\Big(\frac{5^5}{3^5}\Big)^{\frac{4}{5}}$ $=\Big(\frac{5}{3}\Big)^{5\times\frac{4}{5}}$ $=\Big(\frac{5}{3}\Big)^4$ $=\frac{625}{81}$
View full question & answer→Question 311 Mark
Classify the following number as rational or irrational. give reasons to support your answer. $2.356565656...$
Answer$2.356565656...$ is a rational number because it is repeating.
View full question & answer→Question 321 Mark
Simplify: $3^\frac{1}{4}\times5^\frac{1}{4}$
Answer$3^\frac{1}{4}\times5^\frac{1}{4}=(3\times5)^{\frac{1}{4}}=(15)^{\frac{1}{4}}$
View full question & answer→Question 331 Mark
Represent the following rational numbers on the number line: $1.3$
Answer$1.3=\frac{13}{10}=1\frac{3}{10}$
View full question & answer→Question 341 Mark
Is zero a rational number? Justify.
AnswerYes, $0$ is a rational number. $0$ can be expressed in the form of the fraction $\frac{\text{p}}{\text{q}},$ where $p = 0$ and $q$ can be any integer except $0.$
View full question & answer→Question 351 Mark
Without actual division, find the following rational numbers are terminating decimals.
$\frac{13}{80}$
Answer$\frac{13}{80}$
Denominator of $\frac{13}{80}$ is $80 .$
And,
$80=2^4 \times 5$
Therefore, $80$ has no other factors than $2$ and $5 .$
Thus, $\frac{13}{80}$ is a terminating decimal.
View full question & answer→Question 361 Mark
Simplify: $\frac{5^{\frac{6}{7}}}{5^{\frac{2}{3}}}$
Answer$\frac{5^{\frac{6}{7}}}{5^{\frac{2}{3}}}=5^{\big(\frac{6}{7}-\frac{2}{3}\big)}$ $=5^{\big(\frac{18-14}{21}\big)}=5^{\frac{4}{21}}$
View full question & answer→Question 371 Mark
Simplify: $2^\frac{2}{3}\times2^\frac{1}{5}$
Answer$2^\frac{2}{3}\times2^\frac{1}{5}$ $=2^{\frac{2}{3}+\frac{1}{5}}$ $=2^{\frac{10+3}{15}}$ $=2^{\frac{13}{15}}$
View full question & answer→Question 381 Mark
Represent the following rational numbers on the number line: $\frac{8}{3}$
Answer$\frac{8}{3}=2\frac{2}{3}$
View full question & answer→Question 391 Mark
Give an example of two irrational numbers whose: Sum is a rational number.
Answer$2$ irrational numbers with sum a rational number $3-\sqrt{2}$ and $3+\sqrt{2}$
View full question & answer→Question 401 Mark
Simplify: $\frac{8^{\frac{1}{2}}}{8^{\frac{2}{3}}}$
Answer$\frac{8^{\frac{1}{2}}}{8^{\frac{2}{3}}}=8^{\big(\frac{1}{2}-\frac{2}{3}\big)}$ $=8^{\big(\frac{3-4}{6}\big)}=8^{\frac{-1}{6}}$
View full question & answer→Question 411 Mark
Classify the following number as rational or irrational. give reasons to support your answer. $1.23232333...$
Answer$1.23232333...$ is an irrational number because it is a non−terminating, non−repeating decimal.
View full question & answer→Question 421 Mark
Simplify: $(3^4)^{\frac{1}{4}}$
Answer$(3^4)^{\frac{1}{4}}=3^{\big(4\times\frac{1}{4}\big)}=(3)^1=3$
View full question & answer→Question 431 Mark
Simplify $\sqrt[4]{81\text{x}^8\text{y}^4\text{z}^{16}}.$
Answer$\sqrt[4]{81\text{x}^8\text{y}^4\text{z}^{16}}$ $=\sqrt[4]{3^4(\text{x}^2)^4\text{y}^4(\text{z}^4)^4}$ $=\sqrt[4]{(3\text{x}^2\text{y}\text{z}^4)^4}$ $=(3\text{x}^2\text{y}\text{z}^4)^{4\times\frac{1}{4}}$ $=3\text{x}^2\text{y}\text{z}^4$
View full question & answer→Question 441 Mark
Give an example of two irrational numbers whose: Difference is an irrational number.
Answer$2$ irrational numbers with difference an irrational number will be $3-\sqrt{5}$ and $3+\sqrt{5}.$
View full question & answer→Question 451 Mark
Represent the following rational numbers on the number line: $-\frac{23}{6}$
Answer$-\frac{23}{6}=-3\frac{5}{6}$
View full question & answer→Question 461 Mark
Write the rationalising factor of the denominator in $\frac{1}{\sqrt{2}+\sqrt{3}}.$
AnswerThe rationalising factor of the denominator in $\frac{1}{\sqrt{2}+\sqrt{3}}$ is $\big(\sqrt{2}-\sqrt{3}\big).$
View full question & answer→Question 471 Mark
Add: $\Big(\frac{2}{3}\sqrt{7}-\frac{1}{2}\sqrt{2}+6\sqrt{11}\Big)$ and $\Big(\frac{1}{3}\sqrt{7}+\frac{3}{2}\sqrt{2}-\sqrt{11}\Big)$
AnswerWe have: $\Big(\frac{2}{3}\sqrt{7}-\frac{1}{2}\sqrt{2}+6\sqrt{11}\Big)+\Big(\frac{1}{3}\sqrt{7}+\frac{3}{2}\sqrt{2}-\sqrt{11}\Big)$ $=\Big(\frac{2}{3}\sqrt{7}+\frac{1}{3}\sqrt{7}\Big)+\Big(-\frac{1}{2}\sqrt{2}+\frac{3}{2}\sqrt{2}\Big)+\big(6\sqrt{11}-\sqrt{11}\big)$ $=\Big(\frac{2}{3}+\frac{1}{3}\Big)\sqrt{7}+\Big(-\frac{1}{2}+\frac{3}{2}\Big)\sqrt{2}+(6-1)\sqrt{11}$ $=\sqrt{7}+\sqrt{2}+5\sqrt{11}$
View full question & answer→Question 481 Mark
Evaluate $\Big(\frac{81}{49}\Big)^{\frac{-3}{2}}.$
Answer$\Big(\frac{81}{49}\Big)^{\frac{-3}{2}}$ $=\Big(\frac{49}{81}\Big)^{\frac{3}{2}}$ $=\Big(\frac{7^2}{9^2}\Big)^{\frac{3}{2}}$ $=\Big(\frac{7}{9}\Big)^{2\times\frac{3}{2}}$ $=\Big(\frac{7}{9}\Big)^3$ $=\frac{343}{729}$
View full question & answer→Question 491 Mark
Simplify $\sqrt[4]{\sqrt[3]{\text{x}^2}}$ and express the result in the exponential form of $x.$
Answer$\sqrt[4]{\sqrt[3]{\text{x}^2}}$ $=\Big(\sqrt[3]{\text{x}^2}\Big)^\frac{1}{4}$ $=\big(\text{x}^2\big)^{\frac{1}{3}\times\frac{1}{4}}$ $=\text{x}^{2\times\frac{1}{12}}$ $=\text{x}^\frac{1}{6}$
View full question & answer→Question 501 Mark
Simplify $(32)^\frac{1}{5}+(-7)^0+(64)^{\frac{1}{2}}.$
Answer$(32)^\frac{1}{5}+(-7)^0+(64)^{\frac{1}{2}}$ $=(2^5)^{\frac{1}{5}}+1+(8^2)^{\frac{1}{2}}$ $=2^{5\times\frac{1}{5}}+1+8^{2\times\frac{1}{2}}$ $=2+1+8$ $=11$
View full question & answer→Question 511 Mark
If $a=1, b=2$ then find the value of $\left(a^b+b^a\right)^{-1}$.
AnswerGiven, $a = 1$ and $b = 2$
$\therefore(\text{a}^{\text{b}}+\text{b}^{\text{a}})^{-1}$
$=\frac{1}{\text{a}^{\text{b}}+\text{b}^{\text{a}}}$
$=\frac{1}{1^2+2^1}$
$=\frac{1}{1+2}$
$=\frac{1}{3}$
View full question & answer→Question 521 Mark
Simplify $6\sqrt{36}+5\sqrt{12}$
Answer$6\sqrt{3}+5\sqrt{12}$ $=6\sqrt{3}+5\sqrt{4\times3}$ $=6\sqrt{3}+5\times2\sqrt{3}$ $=6\sqrt{3}+10\sqrt{3}$ $=16\sqrt{3}$
View full question & answer→Question 531 Mark
Simplify $\big(2\sqrt{5}+3\sqrt{2}\big)^2.$
Answer$\big(2\sqrt{5}+3\sqrt{2}\big)^2$ $=\big(2\sqrt{5}\big)^2+2\times2\sqrt{5}\times3\sqrt{2}+\big(3\sqrt{2}\big)^2$ $=20+12\sqrt{10}+18$ $=38+12\sqrt{10}$
View full question & answer→Question 541 Mark
Give an example of two irrational numbers whose:
Product is a rational number.
Answer$2$ irrational numbers with product a rational number will be $5+\sqrt{7}$ and $5-\sqrt{7}$
View full question & answer→Question 551 Mark
Classify the following number as rational or irrational. give reasons to support your answer. $\sqrt{361}$
Answer$\sqrt{361}=19$ So, it is rational.
View full question & answer→Question 561 Mark
Evaluate: $\big(25\big)^{\frac{3}{2}}$
Answer$\big(25\big)^{\frac{3}{2}}=(5^2)^{\frac{3}{2}}=5^{\big(2\times\frac{3}{2}\big)}=5^3=125$
View full question & answer→Question 571 Mark
Simplify: $\Big(3^{\frac{1}{3}}\Big)^4$
Answer$\Big(3^{\frac{1}{3}}\Big)^4=3^{\big(\frac{1}{3}\times4\big)}=3^{\frac{4}{3}}$
View full question & answer→Question 581 Mark
Simplify: $\Big({\frac{1}{3^4}}\Big)^{\frac{1}{2}}$
Answer$\Big({\frac{1}{3^4}}\Big)^{\frac{1}{2}}=\big(3^{-4}\big)^{\frac{1}{2}}=3^{\big(-4\times\frac{1}{2}\big)}=3^{-2}$
View full question & answer→Question 591 Mark
If $\sqrt{10}=3.162,$ find the value of $\frac{1}{\sqrt{10}}.$
AnswerGiven, $\sqrt{10}=3.162$ Now, $\frac{1}{\sqrt{10}}=\frac{1}{\sqrt{10}}\times\frac{\sqrt{10}}{\sqrt{10}}=\frac{\sqrt{10}}{\big(\sqrt{10}\big)^2}=\frac{\sqrt{10}}{10}=\frac{3.162}{100}=0.3162$
View full question & answer→Question 601 Mark
Give an example of two irrational numbers whose: Quotient is a rational number.
Answer$2$ irrational numbers with quotient a rational number will be $\sqrt{63}$ and $\sqrt{7}$
View full question & answer→Question 611 Mark
Simplify:
$2^\frac{2}{3}\times2^\frac{1}{3}$
Answer $2^{\frac{2}{3}}\times2^{\frac{1}{3}}$
$=2^{\frac{2}{3}+\frac{1}{3}}$
$=2^{\frac{3}{3}}$
$=2^1$
$=2$
View full question & answer→Question 621 Mark
Evaluate: $\big(8\big)^{-\frac{1}{3}}$
Answer$\big(8\big)^{-\frac{1}{3}}=\frac{1}{\big(8\big)^{\frac{1}{3}}}=\frac{1}{\big(2^3\big)^{\frac{1}{3}}}=\frac{1}{2^{\big(3\times\frac{1}{3}\big)}}$ $=\frac{1}{2^1}=\frac{1}{2}$
View full question & answer→Question 631 Mark
Give an example of two irrational numbers whose: Difference is a rational number.
Answer$2$ irrational numbers with difference is a rational number will be $5+\sqrt{3}$ and $2+\sqrt{3}$
View full question & answer→Question 641 Mark
Evaluate:
$\big(81\big)^{\frac{3}{4}}$
Answer $\big(81\big)^{\frac{3}{4}}=(3^4)^{\frac{3}{4}}=3^{\big(4\times\frac{3}{4}\big)}=3^3=27$
View full question & answer→Question 651 Mark
Classify the following number as rational or irrational. give reasons to support your answer. $\sqrt{21}$
Answer$\sqrt{21}=\sqrt{3}\times\sqrt{7}=4.58257...$ It is an irrational number.
View full question & answer→Question 661 Mark
Find the value of $\frac{21\sqrt{12}}{10\sqrt{27}}.$
Answer$\frac{21\sqrt{12}}{10\sqrt{27}}$ $=\frac{21\sqrt{4\times3}}{10\sqrt{9\times3}}$ $=\frac{21\times2\sqrt{3}}{10\times3\sqrt{3}}$ $=\frac{7}{5}$
View full question & answer→Question 671 Mark
Simplify:
$2^\frac{5}{8}\times3^\frac{5}{8}$
Answer $2^\frac{5}{8}\times3^\frac{5}{8}=(2\times3)^{\frac{5}{8}}=(6)^{\frac{5}{8}}$
View full question & answer→Question 681 Mark
Classify the following number as rational or irrational. give reasons to support your answer. $\sqrt{\frac{3}{81}}$
Answer$\sqrt{\frac{3}{81}}$ $\sqrt{\frac{3}{81}}=\sqrt{\frac{1}{27}}=\frac{1}{3}\sqrt{\frac{1}{3}}$ It is an irrational number.
View full question & answer→Question 691 Mark
Simplify: $7^\frac{5}{6}\times7^\frac{2}{3}$
Answer$\Bigg(7^{\frac{5}{6}}\times7^{\frac{2}{3}}\Bigg)=7^{\big(\frac{5}{6}+\frac{2}{3}\big)}=7^{\big(\frac{5+4}{6}\big)}$ $=7^{\frac{9}{6}}=7^{\frac{3}{2}}$
View full question & answer→Question 701 Mark
Let $x$ be a rational number and $y$ be an irrational number. Is $x+y$ necessarily an irrational number? Give a example in support of your answer.
Answer$x$ be a rational number and $y$ be an irrational number then $x+y$ necessarily will be an irrational number. Example: $5$ is a rational number but $\sqrt{2}$ is irrational.
So, $5+\sqrt{2}$ will be an irrational number.
View full question & answer→Question 711 Mark
Evaluate: $(1^3+2^3+3^3)^{\frac{1}{2}}$
Answer$(1^3+2^3+3^3)^{\frac{1}{2}}$ $=(1+8+27)^{\frac{1}{2}}$ $=(36)^{\frac{1}{2}}$ $=(6^2)^{\frac{1}{2}}$ $=6$
View full question & answer→