# How do you write 625^(3/4) in radical form?

Notice that $625 = {5}^{4}$ hence

$\sqrt[4]{{625}^{3}} = \sqrt[4]{{5}^{12}} = {5}^{3} = 125$

May 28, 2016

If you don't automatically know that ${5}^{4} = 625$, or that ${25}^{2} = 625$, another way to do this is:

$\textcolor{b l u e}{{625}^{\text{3/4}}}$

$= {\left(600 + 25\right)}^{\text{3/4}}$

$= {\left(60 \cdot 10 + 25\right)}^{\text{3/4}}$

$= {\left(120 \cdot 5 + 25\right)}^{\text{3/4}}$

$= {\left(12 \cdot 50 + 25\right)}^{\text{3/4}}$

$= {\left(24 \cdot 25 + 25\right)}^{\text{3/4}}$

$= {\left({25}^{2}\right)}^{\text{3/4}}$

Remember that ${\left({x}^{a}\right)}^{b} = {x}^{a \cdot b}$.

$= {25}^{2 \cdot \text{3/4}}$

$= {25}^{\text{6/4}}$

$= {25}^{\text{3/2}}$

$= {25}^{3 \cdot \text{1/2}}$

$= {\left({5}^{2}\right)}^{3 \cdot \text{1/2}}$

$= {5}^{2 \cdot 3 \cdot \text{1/2}}$

Remember that multiplication is commutative.

$= {5}^{3 \cdot 2 \cdot \text{1/2}}$

$= {\left(5 \cdot 5 \cdot 5\right)}^{2 \cdot \text{1/2}}$

$= {\left(25 \cdot 5\right)}^{1}$

$= \textcolor{b l u e}{125}$