# How do you evaluate 12\sqrt { 2} \div 2\sqrt { 27}?

Jun 11, 2017

It's $\frac{2 \sqrt{6}}{3}$.

#### Explanation:

$\frac{12 \cdot \sqrt{2}}{2 \cdot \sqrt{27}}$

We can't do anything about the $\sqrt{2}$ but we can split the $\sqrt{27}$ into $\sqrt{3}$ and $\sqrt{9}$. We notice that we can simplify the $\sqrt{9}$ into $3$. Now, let's combine everything together:

$\frac{12 \cdot \sqrt{2}}{2 \cdot \sqrt{3} \cdot \sqrt{9}} = \frac{12 \sqrt{2}}{2 \cdot \sqrt{3} \cdot 3} = \frac{12 \sqrt{2}}{6 \sqrt{3}}$

Now what? Well, we see that we can split the final expression into $\frac{12}{6}$ and $\frac{\sqrt{2}}{\sqrt{3}}$. And of course we can simplify the $\frac{12}{6}$ into $2$:

$\frac{12 \sqrt{2}}{6 \sqrt{3}} = \frac{12}{6} \cdot \frac{\sqrt{2}}{\sqrt{3}} = 2 \cdot \frac{\sqrt{2}}{\sqrt{3}}$

But how do we simplify $\frac{\sqrt{2}}{\sqrt{3}}$? There's a trick that we can use and it's called rationalizing the denominator. Essentially we multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ and get this:

$\frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{2} \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{6}}{\sqrt{9}} = \frac{\sqrt{6}}{3}$

There! Now we can go back to the equation and continue working:

$2 \cdot \frac{\sqrt{2}}{\sqrt{3}} = 2 \cdot \frac{\sqrt{6}}{3} = \frac{2 \sqrt{6}}{3}$

Hope this helped!