Question

How do you simplify $81^{\dfrac{3}{4}}$?

Answer 1

Hint:
In the given problem we need to simplify ${{81}^{\dfrac{3}{4}}}$. 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.
Required formula : ${{({{a}^{x}})}^{y}}={{a}^{x\times y}}$

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 $81$ is a square of $9$ .
Therefore it can be written as, ${{({{9}^{2}})}^{\dfrac{3}{4}}}$
Now by the properties of indices. The next step will be,
${{({{9}^{2}})}^{\dfrac{3}{4}}}={{(9)}^{\dfrac{2\times 3}{4}}}$
But we can also say that $9$ is a square of $3$.
Therefore, $9={{(3)}^{2}}$.
Substituting the value of 9 in our equation, we will get
${{({{3}^{2}})}^{\dfrac{2\times 3}{4}}}={{(3)}^{\dfrac{2\times 2\times 3}{4}}}$
Which will further solve into,
$={{(3)}^{3}}$
$=27$

Thus the final answer for the given numerical is $27$.

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.