Question

How do you simplify ${{9}^{\dfrac{5}{2}}}$ ?

Answer 1

Hint: We have been given an exponential function with a constant base 9 and a constant fractional power. We shall first break this fractional power into parts. The power 5 on 9 indicates multiplication of 9, 5 times with itself whereas the power $\dfrac{1}{2}$ on 9 indicates the square root of 9.

Complete step by step solution:
Given that ${{9}^{\dfrac{5}{2}}}$.
The power of this expression, $\dfrac{5}{2}$ can also be written as the multiplication of 5 and $\dfrac{1}{2}$, that is, $\dfrac{5}{2}=5\times \dfrac{1}{2}$
$\Rightarrow {{9}^{\dfrac{5}{2}}}={{9}^{5\times \dfrac{1}{2}}}$
$\Rightarrow {{9}^{\dfrac{5}{2}}}={{9}^{\dfrac{1}{2}\times 5}}$
According to a property of exponential functions, when the powers of any base are being multiplied, then we step-by-step raise the base to each power and then its resultant to the remaining power, that if for any base $a$, if the powers are $b$ and $c$, then it can be expressed as ${{a}^{b\times c}}={{\left( {{a}^{b}} \right)}^{c}}$
$\Rightarrow {{9}^{\dfrac{5}{2}}}={{\left( {{9}^{\dfrac{1}{2}}} \right)}^{5}}$ ……………… Equation (1)
We know that $3\times 3=9$
$\Rightarrow {{3}^{2}}=9$
Here, we shall square root both sides of this equation.
$\Rightarrow \sqrt{{{3}^{2}}}=\sqrt{9}$
The square root symbol, $\sqrt{{}}$ is used when a term is raised to the power $\dfrac{1}{2}$.
$\Rightarrow {{\left( {{3}^{2}} \right)}^{\dfrac{1}{2}}}={{9}^{\dfrac{1}{2}}}$
$\Rightarrow 3={{9}^{\dfrac{1}{2}}}$
Using this in equation (1), we get
$\Rightarrow {{9}^{\dfrac{5}{2}}}={{\left( 3 \right)}^{5}}$
We shall now multiply 3 by itself 5 times.
$\Rightarrow {{9}^{\dfrac{5}{2}}}=3\times 3\times 3\times 3\times 3$
$\Rightarrow {{9}^{\dfrac{5}{2}}}=243$

Therefore, ${{9}^{\dfrac{5}{2}}}$ is simplified to 243.

Note: In order to solve the problems related to the exponents which consist of namely a base and its powers, we must have prior knowledge of the basic properties of exponents. One of the most important properties is that whenever two exponential terms with the same base are multiplied and divided, then their powers are added and subtracted respectively.