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

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

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Joseph Lewis

Explanation:
Breaking $$\displaystyle{9}^{{{\frac{{{3}}}{{{2}}}}}}$$ down into its component parts
$$\displaystyle{\frac{{{3}}}{{{2}}}}$$ is the same as $$\displaystyle{3}\times{\frac{{{1}}}{{{2}}}}$$ which is the same as $$\displaystyle{\frac{{{1}}}{{{2}}}}\times{3}$$
so $$\displaystyle{9}^{{{\frac{{{3}}}{{{2}}}}}}$$ is the same as $$9^{\frac{1}{2}\times3}$$
Write this as: $$\displaystyle{\left({9}^{{{\frac{{{1}}}{{{2}}}}}}\right)}^{{3}}$$
Consider the part of $$\displaystyle{9}^{{{\frac{{{1}}}{{{2}}}}}}$$
This is the same as $$\displaystyle\sqrt{{{9}}}={3}$$
So $$\displaystyle{\left({9}^{{{\frac{{{1}}}{{{2}}}}}}\right)}^{{3}}={\left(\sqrt{{{9}}}\right)}^{{3}}$$
$$\displaystyle{3}^{{3}}={27}$$
However, it true to say that:
$$\displaystyle\sqrt{{{9}}}=\pm{3}$$
and $$\displaystyle{\left(\pm{3}\right)}^{{3}}=\pm{27}$$

Not exactly what you’re looking for?
Timothy Wolff
Simplify the expression.
$$\displaystyle{3}^{{{2}{\left({\frac{{{3}}}{{{2}}}}\right)}}}$$
Cancel the common factor of 2
$$\displaystyle{3}^{{3}}$$
Raise 3 to the power of 3 .
27
Vasquez

$$\begin{array}{}\Rightarrow(9)^{\frac{3}{2}} \\\Rightarrow(3^2)^{\frac{3}{2}} \\\Rightarrow 3^{2\cdot\frac{3}{2}} \\\Rightarrow 3^3=27 \end{array}$$