Question

$\text{If}(\frac{5}{4}{)}^3\times (\frac{12.5}{10}{)}^7\times (\frac{4}{5}{)}^2=(\frac{5}{4}{)}^{3x-1}$ then find the value of $x.$

• A. 6
• B. 1
• C. 3
• D. 4

Answer 1

The correct option is C 3

$(\frac{5}{4}{)}^3\times (\frac{12.5}{10}{)}^7\times (\frac{4}{5}{)}^2=(\frac{5}{4}{)}^{3x-1}$


$\frac{12.5}{10}=\frac{125}{100}=\frac{5}{4}$ and
$(\frac{4}{5}{)}^2=(\frac{5}{4}{)}^{-2}(\because (\frac{a}{b}{)}^n=\left(\frac{b}{a}{)}^{-n}\right)$

$\therefore \text{LHS}=(\frac{5}{4}{)}^3\times (\frac{5}{4}{)}^7\times (\frac{5}{4}{)}^{-2}=(\frac{5}{4}{)}^{3+7+(-2)}(\because a^m\times a^n=a^{m+n})$
$=(\frac{5}{4}{)}^8=\text{RHS}=(\frac{5}{4}{)}^{3x-1}$

i.e., $(\frac{5}{4}{)}^8=(\frac{5}{4}{)}^{3x-1}$
Equal bases and equal powers
$⟹$ Equal exponents
$\therefore 8=3x-1$

$⟹8+1=3x$

$⟹\frac{9}{3}=x$

$⟹x=3$