Question

How do you simplify ${{8}^{\dfrac{3}{4}}}$ ?

Answer 1

Hint: In the given problem we need to simplify ${{8}^{\dfrac{3}{4}}}$. 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.

Formula used:
 ${{({{a}^{x}})}^{y}}={{a}^{x\times y}}$

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 $8$ is a cube of $2$.
Therefore it can be written as, ${{({{2}^{3}})}^{\dfrac{3}{4}}}$
Now by the properties of indices. The next step will be,
${{({{2}^{3}})}^{\dfrac{3}{4}}}={{(2)}^{\dfrac{3\times 3}{4}}}$
Which will further solve into,
${{(2)}^{\dfrac{9}{4}}}$
Now, we know ${{(x)}^{\dfrac{a}{b}}}={{x}^{a}}-{{x}^{b}}$, Hence we can write the given equation as:
${{(2)}^{\dfrac{9}{4}}}={{2}^{9}}-{{2}^{4}}$
$=512-16$
$=496$

Thus the final answer for the given numerical is $496$.

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.