# How do you simplify (3+root3(3))/(root3(9))?

Mar 18, 2017

$\frac{3 + \sqrt[3]{3}}{\sqrt[3]{9}} = \sqrt[3]{3} + \frac{\sqrt[3]{9}}{3}$

#### Explanation:

Simplifying $\frac{3 + \sqrt[3]{3}}{\sqrt[3]{9}}$ means rationalizing the denominator i.e. converting irrational denominator to a rational denominator,

As denominator is $\sqrt[3]{9} = \sqrt[3]{3 \times 3}$, to convert it into a rational number, we need to take everything outside the cube root sign.

However, we could have done so only if we had three $3$'s, but we have only two $3$'s.

Hence, we need to multiply denominator by $\sqrt[3]{3}$, but to keep the expression same, we will have to multiply numerator too by $\sqrt[3]{3}$.

Hence $\frac{3 + \sqrt[3]{3}}{\sqrt[3]{9}}$

= $\frac{3 + \sqrt[3]{3}}{\sqrt[3]{9}} \times \frac{\sqrt[3]{3}}{\sqrt[3]{3}}$

= $\frac{3 \sqrt[3]{3} + \sqrt[3]{3} \times \sqrt[3]{3}}{\sqrt[3]{27}}$

= $\frac{3 \sqrt[3]{3} + \sqrt[3]{9}}{3}$

= $\sqrt[3]{3} + \frac{\sqrt[3]{9}}{3}$