Question

How do you simplify ${{25}^{-\dfrac{1}{2}}}$ ?

Answer 1

Hint: In the given problem we need to simplify ${{25}^{-\dfrac{1}{2}}}$ . These types of expressions are somewhat difficult to evaluate and that’s why they need to be solved carefully. Simplification can be done by breaking the expression with a fractional exponent down into its component parts.

Formula used:
${{x}^{-\dfrac{1}{a}}}=\dfrac{1}{{{x}^{\dfrac{1}{a}}}}$ and ${{({{a}^{x}})}^{y}}={{a}^{x\times y}}$

Complete Step by Step Solution:
The first step in solving such types of problems is to simplify the sum as much as possible. From the given sum, we can say that $25$ is a square of $5$.
Therefore it can be written as, ${{(5)}^{2}}$
Now by the properties of indices. The next step will be,
${{({{5}^{2}})}^{-\dfrac{1}{2}}}={{(5)}^{-\dfrac{2\times 1}{2}}}$
Which will further solve into,
\[{{(5)}^{-1}}\]
Now, we know ${{x}^{-\dfrac{1}{a}}}=\dfrac{1}{{{x}^{\dfrac{1}{a}}}}$ , Hence we can write the given equation as:
\[{{(5)}^{-1}}=\dfrac{1}{5}\]

Thus the final answer for the given numerical is $\dfrac{1}{5}$.

Note:
An exponential expression with a fraction as the exponent is known as an expression with a fractional exponent. These types of expressions are somewhat difficult to evaluate and that’s why they need to be solved carefully.