Question

If we have an expression as ${{\left( 25 \right)}^{x}}={{\left( 125 \right)}^{y}}$ then find $x:y$.
A. $1:1$
B. $2:3$
C. $3:2$
D. $1:3$

Answer 1

Hint: We first need to find the proper bases that we need to use to convert the main equation. We use different theorems of indices to get different power values with the same base. We equate the power values to find the ratio of x and y. We need to convert the given bases in such a way that the simplified form has an equal and basic base.

Complete step-by-step solution
We have been given an equation of indices ${{\left( 25 \right)}^{x}}={{\left( 125 \right)}^{y}}$. The base values are 25, 125.
we try to convert all the base values in the expression of 5.
We have the indices formula of ${{\left( {{x}^{a}} \right)}^{b}}={{x}^{ab}}$.
On the left-hand side, we have ${{\left( 25 \right)}^{x}}={{\left( {{5}^{2}} \right)}^{x}}={{\left( 5 \right)}^{2x}}$.
On the right-hand side we have ${{\left( 125 \right)}^{y}}$. We apply same process and get ${{\left( 125 \right)}^{y}}={{\left( {{5}^{3}} \right)}^{y}}={{\left( 5 \right)}^{3y}}$.
We know that if ${{x}^{a}}={{x}^{b}}\Rightarrow a=b$. Applying this theorem on ${{\left( 5 \right)}^{2x}}={{\left( 5 \right)}^{3y}}$, we get $2x=3y$
We solve the linear equation to get the ratio of x and y.
$\begin{align}
  & 2x=3y \\
 & \Rightarrow \dfrac{x}{y}=\dfrac{3}{2} \\
 & \Rightarrow x:y=3:2 \\
\end{align}$
Therefore, the ratio of x and y is $3:2$. The correct option is C.

Note: Although the norm is to use the basic base rule, to find the solution we could have converted the given equation into some other base other than 5. The general rule is to have the same base. So, instead of 5, we could have taken the base as of 25 or 125. The power would have been fractional in some cases but the important thing is to keep the same base form.