Power, Indices and Surds MCQ Question with Answer

Solution: $16\times 8^{n+2}=2^m$

$\Rightarrow 2^4\times 2^{3n+6}=2^m$

$\Rightarrow 2^{4+3n+6}=2^m$

$\Rightarrow 3n+10 =m$

Solution: $\sqrt[3]{512}=2^x$

$\Rightarrow \sqrt[3]{2^9}=2^x$

$\Rightarrow 2^{\frac{9}{3}}=2^x$

$\Rightarrow x=3$

Solution: $\sqrt{4+\sqrt{x}}=4$

$\Rightarrow 4+\sqrt{x}=16$

$\Rightarrow \sqrt{x}=12$

$\Rightarrow x=144$

Solution: $5^{x+3}\,\,=\,\,\left( 25 \right) ^{3x-4}$

$\Rightarrow 5^{x+3}\,\,=\,\,5^{\begin{array}{c} 6x-8\\ \end{array}}$

$\Rightarrow 6x-8 =\,\,x+3$

$\Rightarrow 5x\,\,=\,\,11$

$\Rightarrow x=\frac{11}{5}$

Solution: $3^{4x-2}\,\,=\,\,729$

$\Rightarrow 3^{4x-2}=3^6$

$\Rightarrow 4x-2=6$

$\Rightarrow x=2$

Solution: $2^{2x-1}\,\,=\,\,\frac{1}{8^{x-3}}$

$\Rightarrow 2^{2x-1}\,\,=\,\,2^{-3\left( x-3 \right)}$

$\Rightarrow 2x-1=-3x+9$

$\Rightarrow 5x=10$

$\Rightarrow x=2$

Solution: $\left( \frac{a}{b} \right) ^{x-1}\,\,=\,\,\left( \frac{b}{a} \right) ^{x-3}$

$\Rightarrow \left( \frac{a}{b} \right) ^{x-1}=\,\,\frac{b^{x-3}}{a^{x-3}}$

$\Rightarrow \left( \frac{a}{b} \right) ^{x-1}=\frac{a^{-\left( x-3 \right)}}{b^{-\left( x-3 \right)}}=\left( \frac{a}{b} \right) ^{-\left( x-3 \right)}$

$\Rightarrow x-1=-x+3$

$\Rightarrow x=2$

Solution: $2^x\times 8^{\frac{1}{5}}=2^{\frac{1}{5}}\,\,$

$2^x\times 2^{\frac{3}{5}}\,\,=2^{\frac{1}{5}}\,\,$

$\Rightarrow 2^{x+\frac{3}{5}}\,\,=2^{\frac{1}{5}}\,\,$

$\Rightarrow x+\frac{3}{5}=\frac{1}{5}$

$\Rightarrow x=\,\,-\frac{2}{5}$

Solution: $2^x-2^{x-1}=4$

$\Rightarrow 2^x-\frac{2^x}{2}=4$

$\Rightarrow \frac{2.2^x-2^x}{2}=4$

$\Rightarrow 2^x\left( 2-1 \right) =8$

$\Rightarrow 2^x=2^3$

$\Rightarrow x=3$

$\Rightarrow x^3=27$

Solution: $2^{x+4}-2^{x-1}=31$

$\Rightarrow 16\times 2^x-\frac{2^x}{2}=31$

$\Rightarrow 2^x\left( 16-\frac{1}{2} \right) =31$

$\Rightarrow 2^x\times \frac{31}{2}=31$

$\Rightarrow 2^x=2^1$

$x=1$