Question

How do you simplify ${9^{\dfrac{3}{2}}}$?

Answer 1

Hint: In the given problem we need to simplify ${9^{\dfrac{3}{2}}}$. 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.

Complete step-by-step answer:
According to the given data, we need to simplify ${9^{\dfrac{3}{2}}}$
For that, first we will be breaking ${9^{\dfrac{3}{2}}}$ down into its component parts
It is known to everyone that, $\dfrac{3}{2}$ is the same as $3 \times \dfrac{1}{2}$ which is the same as $\dfrac{1}{2} \times 3$.
Hence,
 ${9^{\dfrac{3}{2}}}$ is the same as ${9^{3 \times \dfrac{1}{2}}}$
Hence, we write it down as: ${({9^{\dfrac{1}{2}}})^3}$
First we will consider the part of ${9^{\dfrac{1}{2}}}$ and try to evaluate it.
This is the same as ${9^{\dfrac{1}{2}}} = \sqrt 9 $
On solving we get,
$\sqrt 9 = 3$
 So ${({9^{\dfrac{1}{2}}})^3}$$ = {(\sqrt 9 )^3}$
Further solving the expression we get,
${({9^{\dfrac{1}{2}}})^3} = {3^3} = 27$
 Therefore, ${9^{\dfrac{3}{2}}} = 27$.

Additional Information:
However, it is important to consider the other possibility too which is,
$ \Rightarrow \sqrt 9 = \pm 3$
Therefore,
$ \Rightarrow ( \pm {3^3}) = \pm 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 we need to be solved carefully.