Question

If ${10^{2y}} = 25$, then ${10^{ - y}}$ is equal to
A) $ - \dfrac{1}{5}$
B) $\dfrac{1}{{625}}$
C) $\dfrac{1}{{50}}$
D) $\dfrac{1}{{25}}$
E) $\dfrac{1}{5}$

Answer 1

Hint:
To find the value ${10^{ - y}}$, with the help of exponent rules, find the value of ${10^y}$. Find the reciprocal of the value, which is resultant of the value ${10^{ - y}}$.(Usually for these types of questions we need to be aware of exponent rules).

Complete step by step solution:
Given that, ${10^{2y}} = 25$
With the help of exponent rule, ${10^{2y}}$can be written as ${10^y} \times {10^y}$which is ${({10^y})^2}$. This is called the power of a power property and says that to find a power of a power you just have to multiply the exponents.
So, now we have
$
   \Rightarrow {({10^y})^2} = 25 \\
   \Rightarrow {({10^y})^2} = {5^2} \\
    \\
 $
Apply square root on both the sides, we have
$
   \Rightarrow {({10^y})^2} = {5^2} \\
\text{On taking square root both sides} \\
   \Rightarrow \sqrt {{{\left( {{{10}^y}} \right)}^2}} = \sqrt {{5^2}} \\
   \Rightarrow {10^y} = 5 \\
   \Rightarrow \dfrac{1}{{{{10}^y}}} = \dfrac{1}{5} \\
   \Rightarrow {10^{ - y}} = \dfrac{1}{5} \\
 $

So, the value of ${10^{ - y}}$ is $\dfrac{1}{5}$,and the correct option is E.

Note:
Aware of what type of exponent rule to be applied on different conditions of the exponents. Properties of exponents are to be understood to apply these properties in these types of questions.