# How do you simplify root3(y^6)/(root3(27y)*root3(y^11))?

Feb 22, 2017

$\frac{1}{3 {y}^{2}}$

#### Explanation:

Because all the terms are cube rooted (and the expression contains no addition or subtraction), the cube root can be moved to the outside of the expression: $\sqrt{{y}^{6} / \left(27 y . {y}^{11}\right)}$

We can simplify the denominator using the fact that ${a}^{n} . {a}^{m} = {a}^{n + m}$
Both terms in the denominator are $y$ raised to a power, so when we multiply them we add the indices to get:

$\sqrt{{y}^{6} / \left(27 {y}^{12}\right)}$

We can now divide the top and the bottom of the fraction by ${y}^{6}$ giving us $1$ in the numerator, as ${y}^{6} / {y}^{6} = 1$

For the denominator we use ${a}^{n} / {a}^{m} = {a}^{n - m}$ to get: $27 {y}^{12} / {y}^{6} = 27 {y}^{12 - 6} = 27 {y}^{6}$

So the expression becomes: $\sqrt{\frac{1}{27 {y}^{6}}}$

The cube root of $27$ is $3$.

$\sqrt{x}$ is equivalent to ${x}^{\frac{1}{3}}$, so $\sqrt{{y}^{6}} = {y}^{\frac{6}{3}} = {y}^{2}$

Therefore the expression fully simplifies to: $\frac{1}{3 {y}^{2}}$