Question 13 Marks
Evaluate: $\frac{x^{5+n}\left(x^2\right)^{3 n+1}}{x^{7 n-2}}$
Answer$ \frac{x^{5+n} \times\left(x^2\right)^{3 n+1}}{x^{7 n-2}}$
$ =\frac{x^{5+n} \times x^{2(3 n+1)}}{x^{7 n-2}}$
$ =\frac{x^{5+n} \times x^{6 n+2}}{x^{7 n-2}}$
$ =x^{5+n+6 n+2-7 n+2}$
$ =x^9$
View full question & answer→Question 23 Marks
Show that : $\left(\frac{x^a}{x^{-b}}\right)^{a-b} \cdot\left(\frac{x^b}{x^{-c}}\right)^{b-c} \cdot\left(\frac{x^c}{x^{-a}}\right)^{c-a}=1$
Answer$\text{L. H. S}$.
$\left(\frac{x^a}{x^{-b}}\right)^{a-b} \cdot\left(\frac{x^b}{x^{-c}}\right)^{b-c} \cdot\left(\frac{x^c}{x^{-a}}\right)^{c-a}=1$
$ =\left(x^{a+b}\right)^{a-b} \cdot\left(x^{b+c}\right)^{b-c} \cdot\left(x^{c+a}\right)^{c-a}$
$ =x^{a^2-b^2} \cdot x^{b^2-c^2} \cdot x^{c^2-a^2}$
$ =x^{a^2-b^2+b^2-c^2+c^2-a^2}$
$ =x^0$
$ =1=\text { R.H.S. }$
View full question & answer→Question 33 Marks
Simplify and express as positive indice: $\left(a^{-2} b\right)^{\frac{1}{2}} \times\left(a b^{-3}\right)^{\frac{1}{3}}$
Answer$ \left(a^{-2} b\right)^{\frac{1}{2}} \times\left(a b^{-3}\right)^{\frac{1}{3}}$
$ =\left(a^{-2 \times \frac{1}{2}} \cdot b^{\frac{1}{2}}\right) \times\left(a^{\frac{1}{3}} b^{-3 \times \frac{1}{3}}\right)$
$ =a^{-1} b^{\frac{1}{2}} \times a^{\frac{1}{3}} b^{-1}$
$ =a^{-1+\frac{1}{3}} b^{\frac{1}{2}-1}$
$ =a^{\frac{-2}{3}} b^{\frac{-1}{2}}$
$ =\frac{1}{a^{\frac{2}{3}} b \frac{1}{2}}$
View full question & answer→Question 43 Marks
Simplify and express as positive indice: $\left[\frac{32 x^{-5}}{243 y^{-5}}\right]^{-\frac{1}{5}}$
Answer$\left[\frac{32 x^{-5}}{243 y^{-5}}\right]^{-\frac{1}{5}}$
$=\left[\frac{2^5 \times x^{-5}}{3^5 y^{-5}}\right]^{-\frac{1}{5}}$
$ =\frac{2^{5 \times \frac{-1}{5}} \cdot x^{-5 \times \frac{-1}{5}}}{3^{5 \times \frac{-1}{5}} \cdot y^{-5} \times \frac{-1}{5}}$
$ =\frac{2^{-1} x^{+1}}{3^{-1} y^{+1}}$
$ =\frac{3 x}{2 y}$
View full question & answer→Question 53 Marks
simplify and express as positive indice: $\left(a^{-2} b\right)^{-2} \cdot(a b)^{-3}$
Answer$ \left(a^{-2} b\right)^{-2} \cdot(a b)^{-3}$
$ =\left(a^{-2 x-2 \cdot b^{-2}}\right) \cdot\left(a^{-3} \cdot b^{-3}\right)$
$ =a^{+4} \cdot b^{-2} \cdot a^{-3} \cdot b^{-3}$
$ =a^{4-3} \cdot b^{-2-3}$
$ =a b^{-5}$
$ =\frac{a}{b^5}$
View full question & answer→Question 63 Marks
Simplify: $\left(x^{a+b}\right)^{a-b} \cdot\left(x^{b+c}\right)^{b-c} \cdot\left(x^{c+a}\right)^{c-a}$
Answer$ \left(x^{a+b}\right)^{a-b} \cdot\left(x^{b+c}\right)^{b-c} \cdot\left(x^{c+a}\right)^{c-a}$
$ =x^{(a+b)(a-b)} \cdot x^{(b+c)(b-c)} \cdot x^{(c+a)(c-a)}$
$ =x^{a^2-b^2} \cdot x^{b^2-c^2} \cdot x^{c^2-a^2}$
$ =x^{a^2-b^2+b^2-c^2+c^2-a^2}$
$ =x^0$
$ =1$
View full question & answer→Question 73 Marks
Simplify: $\left(-2 x^{\frac{2}{3}} y^{-\frac{3}{2}}\right)^6$
Answer$ \left(-2 x^{\frac{2}{3}} y^{-\frac{3}{2}}\right)^6$
$=(-2)^6 \cdot x^{\frac{2}{3} \times 6} \cdot y^{-\frac{3}{2} \times 6}$
$ =64 x^4 y^{-9}$
$ =\frac{64 x^4}{y^9}$
$ =64 x^4 y^{-9}$
View full question & answer→Question 83 Marks
Simplify: $\left(27 x^{-3} y^6\right)^{\frac{2}{3}}$
Answer$ \left(27 x ^{-3} y ^6\right)^{\frac{2}{3}}$
$=(27)^{\frac{2}{3}} \cdot x ^{-3 \times \frac{2}{3}} \cdot y ^{6 \times \frac{2}{3}}$
$ =(3 \times 3 \times 3)^{\frac{2}{3}} \cdot x ^{-2} \cdot y ^4$
$ =\left[(3 \times 3 \times 3)^{\frac{1}{3}}\right]^2 \cdot x ^{-2} \cdot y ^4$
$ =3^2 \cdot x ^{-2} y ^4$
$ =9 x ^{-2} y ^4$
$ =\frac{9 y ^4}{ x ^2}=9 x ^{-2} y ^4$
View full question & answer→Question 93 Marks
Simplify: $\left(2 x^2 y^{-3}\right)^{-2}$
Answer$ \left(2 x^2 y^{-3}\right)^{-2}$
$=2^{-2} \cdot x^{2 \times-2} \cdot y^{-3 \times-2}$
$ =\frac{1}{2^2} x^{-4} \cdot y^6$
$ =\frac{1}{4} \times \frac{y^6}{x^4}$
$ =\frac{y^6}{4 x^4}=\frac{1}{4} \cdot y^6 \cdot x^{-4}$
View full question & answer→Question 103 Marks
Simplify: $\left(125 x^{-3}\right)^{\frac{1}{3}}$
Answer$\left(125 x^{-3}\right)^{\frac{1}{3}}$
$ =(125)^{\frac{1}{3}} \cdot x ^{-3 \times \frac{1}{3}} \ldots\left(\because\left( a ^{ m }\right)^{ n }= a ^{ m \times n } \text { and } a ^{ m } \times a ^{ n }= a ^{ m \times n }\right)$
$ =(5 \times 5 \times 5)^{\frac{1}{3}} \cdot x ^{-1}$
$ =\left(5^3\right)^{\frac{1}{3}} \cdot x^{-1}$
$ =\left(5^{\not \not}\right)^{\frac{1}{\not}} \cdot x^{-1}$
$ =5 x^{-1}$
View full question & answer→Question 113 Marks
Simplify: $\left(2^{-3}-2^{-4}\right)\left(2^{-3}+2^{-4}\right)$
Answer$ \left(2^{-3}-2^{-4}\right)\left(2^{-3}+2^{-4}\right)$
$ =\left(2^{-3}\right)^2-\left(2^{-4}\right)^2$
$ \left\{\because( a - b )( a + b )= a ^2- b ^2\right\}$
$ =2^{-6}-2^{-8}=\frac{1}{2^6}-\frac{1}{2^8}$
$ =\frac{1}{64}-\frac{1}{256}$
$ =\frac{4-1}{256}=\frac{3}{256}$
View full question & answer→Question 123 Marks
Simplify: $8^{\frac{4}{3}}+25^{\frac{3}{2}}-\left(\frac{1}{27}\right)^{-\frac{2}{3}}$
Answer$ 8^{\frac{4}{3}}+25^{\frac{3}{2}}-\left(\frac{1}{27}\right)^{-\frac{2}{3}}$
$ =\left(2^3\right)^{\frac{4}{3}}+\left(5^2\right)^{\frac{3}{2}}-\left(\frac{1}{3^3}\right)^{-\frac{2}{3}}$
$ =2^{3 \times \frac{4}{3}}+5^{2 \times \frac{3}{2}}-\frac{1}{3^{3 \times\left(\frac{-2}{3}\right)}}$
$ =2^4+5^3-\frac{1}{3^{-2}}$
$ =16+125-3^2$
$ =141-9=132$
$ =2^{3 \times \frac{4}{3}}+5^{2 \times \frac{3}{2}}-\frac{1}{3^{3 \times\left(-\frac{2}{3}\right)}}$
$ =2^4+5^3-\frac{1}{3^{-2}}$
$ =16+125-3^2$
$ =141-9$
$=132$
View full question & answer→Question 133 Marks
Simplify: $\frac{x^{2 n+7} \cdot\left(x^2\right)^{3 n+2}}{x^{4(2 n+3)}}$
Answer$ \frac{x^{2 n+7} \cdot\left(x^2\right)^{3 n+2}}{x^{4(2 n+3)}}$
Given expression$=\frac{x^{2 n+7} \cdot x^{6 n+4}}{x^{8 n+12}}$
$ =\frac{x^{2 n+7+6 n+4}}{x^{8 n+12}}=\frac{x^{8 n+11}}{x^{8 n+12}}$
$ =x^{8 n+11-8 n-12}=x^{-1}$
$ =\frac{1}{x}$
View full question & answer→Question 143 Marks
Simplify: $\frac{a^{2 n+3} \cdot a^{(2 n+1)(n+2)}}{\left(a^3\right)^{2 n+1} \cdot a^{n(2 n+1)}}$
Answer$ \frac{a^{2 n+3} \cdot a^{(2 n+1)(n+2)}}{\left(a^3\right)^{2 n+1} \cdot a^{n(2 n+1)}}$
Given expression $=\frac{a^{2 n+3} \cdot a^{\left(2 n^2+4 n+n+2\right)}}{a^{6 n+3} \cdot a^{2 n^2+n}}$
$ =\frac{a^{2 n+3+2 n^2+5 n+2}}{a^{6 n+3+2 n^2+n}}$
$=\frac{a^{2 n^2+7 n+5}}{a^{2 n^2+7 n+3}}$
$ =\frac{a^{\left(2 n^2+7 n+3\right)+2}}{a^{2 n^2+7 n+3}}$
$=a^2$
View full question & answer→Question 153 Marks
Find the value of $n$, when: $12^{-5} \times 12^{2 n +1}=12^{13} \div 12^7 C$
Answer$ 12^{-5} \times 12^{2 n +1}=12^{13} \div 12^7$
$12^{-5+2 n +1}=\frac{12^{13}}{12^7}$
$12^{2 n -4}=12^{13-7}$
$12^{2 n -4}=12^6 $
Comparing both sides, we get
$ 2 n -4=6$
$\Rightarrow 2 n =6+4$
$\Rightarrow 2 n =10$
$\Rightarrow n =5 $
View full question & answer→Question 163 Marks
Prove that: $(m+n)^{-1}\left(m^{-1}+n^{-1}\right)=(m n)^{-1}$
Answer$\text { L.H.S. }( m + n )^{-1}\left( m ^{-1}+ n ^{-1}\right)$
$ =\frac{1}{ m + n }\left(\frac{1}{ m }+\frac{1}{ n }\right)=\frac{1}{ m + n } \cdot \frac{ n + m }{ mn }=\frac{1}{ nn }$
$ =( mn )^{-1}$
$ =\text { R.H.S. }$
Hence Proved.
View full question & answer→Question 173 Marks
Compute: $(625)^{-\frac{3}{4}}$
Answer$(625)^{-\frac{3}{4}} $
$=(5 \times 5 \times 5 \times 5)^{-\frac{3}{4}}$
$ =\left(5^4\right)^{-\frac{3}{4}}=5^{4 \times-\frac{3}{4}}$
$ =5^{-3}=\frac{1}{5^3}$
$ =\frac{1}{5 \times 5 \times 5}$
$ =\frac{1}{125}$
View full question & answer→Question 183 Marks
Compute: $(27)^{\frac{2}{3}} \div\left(\frac{81}{16}\right)^{-\frac{1}{4}}$
Answer$ (27)^{\frac{2}{3}} \div\left(\frac{81}{16}\right)^{-\frac{1}{4}}$
$=\left(3^3\right)^{\frac{2}{3}} \div\left(\frac{3^4}{2^4}\right)^{-\frac{1}{4}}$
$ =3^{3 \times \frac{2}{3}} \div \frac{3^{-\frac{1}{4} \times 4}}{2^{-\frac{1}{4} \times 4}}=3^2 \div \frac{3^{-1}}{2^{-1}}$
$ =3^2 \times \frac{2^{-1}}{3^{-1}}$
$ =3^{2+1} \times 2^{-1}=3^3 \times \frac{1}{2^{+}} 1$
$ =\frac{3 \times 3 \times 3}{2}=\frac{27}{2}=13 \frac{1}{2}$
View full question & answer→Question 193 Marks
Compute: $\left(\frac{27}{64}\right)^{-\frac{2}{3}} $
Answer$\left(\frac{27}{64}\right)^{-\frac{2}{3}} $
$=\left[\frac{\left(3^3\right)}{\left(4^3\right)}\right]^{-\frac{2}{3}}$
$ =\frac{3^{3 \times-\frac{2}{3}}}{4^{3 \times \frac{2}{3}}}=\frac{3^{-2}}{4^{-2}}$
$ =\frac{4^2}{3^2}=\frac{4 \times 4}{3 \times 3}=\frac{16}{9}$
$ =1 \frac{7}{9}$
View full question & answer→Question 203 Marks
Evaluate: $\frac{a^{2 n+1} \times a^{(2 n+1)(2 n-1)}}{a^{n(4 n-1)} \times\left(a^2\right)^{2 n+3}}$
Answer$ \frac{a^{2 n+1} \times a^{(2 n+1)(2 n-1)}}{a^{n(4 n-1)} \times\left(a^2\right)^{2 n+3}}$
$ =\frac{a^{2 n+1} \times a^{(2 n)^2-(1)^2}}{a^{4 n^2-n} \times a^{2(2 n+3)}}$
$ =\frac{a^{2 n+1} \times a^{4 n^2-1}}{a^{4 n^2-n} \times a^{4 n+6}}$
$ =a^{2 n+1+4 n^2-1-4 n^2+n-4 n-6}$
$ =a^{-n-6}$
$ =a^{-(-n+6)}$
$ =\frac{1}{a^{n+6}}$
View full question & answer→Question 213 Marks
Compute: $(12)^{-2} \times 3^3 $
Answer$(12)^{-2} \times 3^3$
$ =(2 \times 2 \times 3)^{-2} \times 3^3$
$ =\left(2^2 \times 3\right)^{-2} \times 3^3$
$ =2^{-2 \times 2} \times 3^{-2} \times 3^3$
$ =2^{-4} \times 3^{-2+3}$
$ =2^{-4} \times 3^1$
$ =\frac{3}{2^4}=\frac{3}{2 \times 2 \times 2 \times 2}$
$ =\frac{3}{16}$
View full question & answer→Question 223 Marks
Compute: $\left(\frac{56}{28}\right)^0 \div\left(\frac{2}{5}\right)^3 \times \frac{16}{25}$
Answer$ \left(\frac{56}{28}\right)^0 \div\left(\frac{2}{5}\right)^3 \times \frac{16}{25}$
$ =1 \div \frac{2^3}{5^3} \times \frac{2 \times 2 \times 2 \times 2}{5 \times 5}$
$ {\left[\because\left(\frac{56}{28}\right)^0=1\right]}$
$ =1 \times \frac{5^3}{2^3} \times \frac{2^4}{5^2}=5^{3-2} \times 2^{4-3}$
$ =5^1 \times 2^1=10$
View full question & answer→Question 233 Marks
Compute: $\left(\frac{2}{3}\right)^{-4} \times\left(\frac{27}{8}\right)^{-2}$
Answer$ \left(\frac{2}{3}\right)^{-4} \times\left(\frac{27}{8}\right)^{-2}$
$=\left(\frac{2}{3}\right)^{-4} \times\left(\frac{3^3}{2^3}\right)^{-2}$
$ =\frac{2^{-4}}{3^{-4}} \times \frac{3^{-6}}{2^{-6}}=\frac{2^{-4}}{2^{-6}} \times \frac{3^{-6}}{3^{-4}}$
$ =2^{-4+6} \times \frac{1}{3^{-4+6}}=\frac{2^2}{3^2}$
$ =\frac{4}{9}$
View full question & answer→Question 243 Marks
Find $x$, if $9 \times 3^x = (27) ^{2x-3}$
Answer$ 9 \times 3^x=(27)^{2 x-3} $
$ 3^2 \times 3^x=(3 \times 3 \times 3)^{2 x-3} $
$ \Rightarrow 3^{x+2}=(3)^{3(2 x-3)} $
$ \Rightarrow 3^{x+2}=(3)^{6 x-9}$
Since, bases are same, compare them,
$ x+2=6 x-9 $
$ 6 x-x=9+2 $
$ \Rightarrow 5 x=11 $
$ \Rightarrow x=\frac{11}{5} \text { i.e. } 2 \frac{1}{5}$
View full question & answer→Question 253 Marks
If $1125 = 3^m x 5^n,$ find $m$ and $n$
Answer$1125 = 3^2 x 5^3$
The factor of $1125$ are $3\times 3 \times 5 \times 5 \times 5$
| $3$ |
$1125$ |
| $3$ |
$375$ |
| $5$ |
$125$ |
| $5$ |
$25$ |
| $5$ |
$5$ |
| |
$1$ |
$\therefore 1125=3 \times 3 \times 5 \times 5 \times 5$
Now comparing, $3^2 \times 5^3=3^m \times 5^n$
$\therefore m =2, n =3$ View full question & answer→Question 263 Marks
Evaluate: $\left[\left(\frac{1}{4}\right)^{-3}-\left(\frac{1}{3}\right)^{-3}\right] \div\left(\frac{1}{6}\right)^{-3}$
Answer$ {\left[\left(\frac{1}{4}\right)^{-3}-\left(\frac{1}{3}\right)^{-3}\right] \div\left(\frac{1}{6}\right)^{-3}}$
$ =\left[\left(\frac{4}{1}\right)^3-\left(\frac{3}{1}\right)^3\right] \div\left(\frac{6}{1}\right)^3$
$ =\left(\frac{4}{1} \times \frac{4}{1} \times \frac{4}{1}-\frac{3}{1} \times \frac{3}{1} \times \frac{3}{1}\right) \div\left(\frac{6}{1}\right)^3$
$ =64-27 \times\left(\frac{1}{6} \times \frac{1}{6} \times \frac{1}{6}\right)$
$ =37 \times \frac{1}{216}=\frac{37}{216}$
View full question & answer→