Question

Find the value of $x$ so that;
$\left(i\right){\left(\frac{3}{4}\right)}^{2x+1}={\left({\left(\frac{3}{4}\right)}^3\right)}^3$
$\left(ii\right){\left(\frac{2}{5}\right)}^3\times {\left(\frac{2}{5}\right)}^6={\left(\frac{2}{5}\right)}^{3x}$
$\left(iii\right){(-\frac{1}{5})}^{20}÷{(-\frac{1}{5})}^{15}={(-\frac{1}{5})}^{5x}$
$\left(iv\right)\frac{1}{16}\times {\left(\frac{1}{2}\right)}^2={\left(\frac{1}{2}\right)}^{3(x-2)}$

Answer 1

$\left(i\right){\left(\frac{3}{4}\right)}^{2x+1}={\left({\left(\frac{3}{4}\right)}^3\right)}^3$

$\Rightarrow (\frac{3}{4}{)}^{2x+1}=({\frac{3}{4}}^9)$ [$\because$ Power of power rule: $(a^m{)}^n=a^{m\times n}$]

We know that, when the bases are equal, then their powers also equal.

$\Rightarrow 2x+1=9$

$\Rightarrow 2x=9-1$

$\Rightarrow 2x=8$

$\Rightarrow x=\frac{8}{2}$

$\Rightarrow x=4$

$\left(ii\right){\left(\frac{2}{5}\right)}^3\times {\left(\frac{2}{5}\right)}^6={\left(\frac{2}{5}\right)}^{3x}$

$\Rightarrow {\left(\frac{2}{5}\right)}^{3+6}={\left(\frac{2}{5}\right)}^{3x}$ [$\because$ Product rule:$a^m\times a^n=a^{m+n}$]

$\Rightarrow {\left(\frac{2}{5}\right)}^9={\left(\frac{2}{5}\right)}^{3x}$

We know that, when the bases are equal, then their powers also equal.

$\Rightarrow 9=3x$

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

$\Rightarrow 3=x$

$\left(iii\right){(-\frac{1}{5})}^{20}÷{(-\frac{1}{5})}^{15}={(-\frac{1}{5})}^{5x}$

$\Rightarrow {(-\frac{1}{5})}^{20-15}={(-\frac{1}{5})}^{5x}[$ $\because$ Quotient rule:$a^m÷a^n=a^{m-n}$]

$\Rightarrow {(-\frac{1}{5})}^5={(-\frac{1}{5})}^{5x}$

We know that, when the bases are equal, then their powers also equal.

$\Rightarrow 5x=5$

$\Rightarrow x=1$

$\left(iv\right)\frac{1}{16}\times {\left(\frac{1}{2}\right)}^2={\left(\frac{1}{2}\right)}^{3(x-2)}$

$\Rightarrow (\frac{1}{2}{)}^4\times {\left(\frac{1}{2}\right)}^2={\left(\frac{1}{2}\right)}^{3(x-2)}$

$\Rightarrow (\frac{1}{2}{)}^{4+2}=(\frac{1}{2}{)}^{3(x-2)}$ [ $\because$ $a^m\times a^n=a^{m+n}$]

We know that, when the bases are equal, then their powers also equal.

$\Rightarrow 3(x-2)=6$

$\Rightarrow x-2=2$

$\Rightarrow x=4$