How do you simplify sqrt(108x^5y^6)?

Apr 19, 2018

If you only want integer powers, then the following is simplified:

$\implies 6 {y}^{3} \sqrt{3 {x}^{5}}$

If you are okay with fractional powers, then you could equivalently write:

$\implies \left(6 \sqrt{3}\right) {x}^{\frac{5}{2}} {y}^{3}$

Explanation:

We are given:

$\sqrt{108 {x}^{5} {y}^{6}}$

First, let's deal with the constant term $108$. We look for factors of $108$ that are perfect squares, as this will allow us to pull it out of the radical.

Factors of $108 = \left\{\textcolor{b l u e}{1} , 2 , 3 , \textcolor{b l u e}{4} , 6 , \textcolor{b l u e}{9} , 12 , 18 , 27 , \textcolor{b l u e}{36} , 54 , 108\right\}$.

The factors in $\textcolor{b l u e}{\text{blue}}$ are perfect squares. To simplify the most, we want the largest one. So we choose $36$, which multiplied by $3$ gives $108$.

$\sqrt{\left(36\right) \left(3\right) {x}^{5} {y}^{6}}$

$\implies \sqrt{\left({6}^{2}\right) \left(3\right) {x}^{5} {y}^{6}}$

$\implies 6 \sqrt{3 {x}^{5} {y}^{6}}$

We can also simplify the variable powers. Taking the square root is equivalent to raising to the $\frac{1}{2}$ power. If you only want integer powers, then the following is simplified:

$6 {y}^{\frac{6}{2}} \sqrt{3 {x}^{5}}$

$\implies 6 {y}^{3} \sqrt{3 {x}^{5}}$

If you are okay with fractional powers, then you could equivalently write:

$\implies \left(6 \sqrt{3}\right) {x}^{\frac{5}{2}} {y}^{3}$