# How do you simplify (3) [square root of (3/7)] -(5) (square root of 84)?

Oct 8, 2015

$- 9 \sqrt{21}$

#### Explanation:

Start by writing down your starting expression

$3 \sqrt{\frac{3}{7}} - 5 \sqrt{84} = 3 \cdot \frac{\sqrt{3}}{\sqrt{7}} - 5 \sqrt{84}$

Thee first thing to do here is rationalize the denominator of the first term by multiplying by $1 = \frac{\sqrt{7}}{\sqrt{7}}$

This will get you

$3 \cdot \frac{\sqrt{3}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} = 3 \cdot \frac{\sqrt{21}}{\sqrt{7} \cdot \sqrt{7}} = \frac{3}{7} \cdot \sqrt{21}$

Now focus on the second radical term. Notice that you can write $84$ as

$84 = 2 \cdot 42 = 2 \cdot 2 \cdot 21 = {2}^{2} \cdot 21$

The expression will thus be

$\frac{3}{7} \cdot \sqrt{21} - 5 \cdot \sqrt{{2}^{2} \cdot 21}$

$\frac{3}{7} \cdot \sqrt{21} - 5 \cdot 2 \sqrt{21}$

This will be equal to

$\sqrt{21} \cdot \left(\frac{3}{7} - 10\right) = \sqrt{21} \cdot \left(\frac{3 - 70}{7}\right) = \sqrt{21} \cdot \frac{\left(- 63\right)}{7}$

Finally, this expression can be simplified to

$- \frac{\left(63\right)}{7} \sqrt{21} = \textcolor{g r e e n}{- 9 \sqrt{21}}$