# How do you simplify ( sqrt32)/ (5 sqrt 14)?

Feb 26, 2016

$\frac{4 \sqrt{7}}{35}$

#### Explanation:

You have to try and spot common values that can be cancelled out.

Both 32 and 14 are even so 2 has to be a common factor giving:

$\text{ } \frac{\sqrt{2 \times 16}}{5 \sqrt{2 \times 7}}$

7 is a prime number so the denominator can not be broken down any further

However, 16 in the numerator can be broken down into ${4}^{2}$ so we now have:

$\text{ } \frac{\sqrt{2 \times {4}^{2}}}{5 \sqrt{2 \times 7}}$

$\text{ } \frac{1}{5} \times \frac{\sqrt{2}}{\sqrt{2}} \times \frac{\sqrt{{4}^{2}}}{\sqrt{7}}$

$\text{ " 1/5xx 1 xx 4/sqrt(7)" "=" } \frac{4}{5 \sqrt{7}}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I forgot that standard practice is that you try to remove any roots from the denominator!

Multiply by 1 but in the form of $\frac{\sqrt{7}}{\sqrt{7}}$ giving:

$\text{ "4/(5sqrt(7))xx sqrt(7)/sqrt(7)" "=" "(4sqrt(7))/(5xx7)" "=" } \frac{4 \sqrt{7}}{35}$

Feb 26, 2016

$\frac{\sqrt{32}}{5 \sqrt{14}} = \frac{4 \sqrt{7}}{35}$

#### Explanation:

$\frac{\sqrt{32}}{5 \sqrt{14}}$

Simplify $\sqrt{32}$.

$\sqrt{2 \times 2 \times 2 \times 2 \times 2} =$

$\sqrt{{2}^{2} \times {2}^{2} \times 2} =$

$4 \sqrt{2}$

Add this back into the expression.

$\frac{4 \sqrt{2}}{5 \sqrt{14}}$

Rationalize the denominator by multiplying both the numerator and denominator by $\sqrt{14}$.

$\frac{4 \sqrt{2}}{5 \sqrt{14}} \times \frac{\sqrt{14}}{\sqrt{14}} =$

$\frac{4 \sqrt{2} \sqrt{14}}{5 \times 14}$

Simplify the numerator by multiplying the square roots.

$\frac{4 \sqrt{28}}{5 \times 14}$

Simplify the square root by factoring.

$\frac{4 \sqrt{28}}{5 \times 14} = \frac{4 \sqrt{2 \times 2 \times 7}}{5 \times 14} = \frac{4 \times 2 \sqrt{7}}{5 \times 14} = \frac{8 \sqrt{7}}{5 \times 14}$

Simplify the denominator.

$\frac{8 \sqrt{7}}{70}$

Simplify the expression.

$\frac{4 \sqrt{7}}{35}$