# How do you simplify the cubed root of 150?

Mar 27, 2016

$\textcolor{g r e e n}{\text{Depending on the interpretation of the question:}}$

$\textcolor{b l u e}{750 \sqrt{6} \text{ ") color(brown)(" As an exact value (decimals are not always exact)}}$

#### Explanation:

Braking the question down into its component parts:

Cubecolor(red)("d") -> (?)^3
root of 150$\to {\left(\sqrt{150}\right)}^{3}$

If the question had read: "cube root of 150 " we would have $\sqrt[3]{150}$

Although, it is not common for people just say "root" for "square root". However, I have come across it being used in that way.

Thus there could be a contradiction in meaning of the question!
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Solution for 'cubed root'}}$

Consider the prime factor tree of 150

We observe that 150 can be broken down to $2 \times 3 \times {5}^{2}$

So $\sqrt{150} = \sqrt{2 \times 3 \times {5}^{2}} = 5 \sqrt{6}$ giving

${\left(\sqrt{150}\right)}^{3} = {\left(5 \sqrt{6}\right)}^{3}$

This gives us

$5 \sqrt{6} \times 5 \sqrt{6} \times 5 \sqrt{6}$

${5}^{3} \times {\left(\sqrt{6}\right)}^{2} \times \sqrt{6}$

$125 \times 6 \times \sqrt{6}$

$\textcolor{b l u e}{750 \sqrt{6} \text{ ")color(brown)(" As an exact value (decimals are not always exact}}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{b l u e}{\text{Solution for 'cube root'}}$

$\sqrt[3]{150}$

George is correct. This can not be simplified any further.