# How do you simplify (7sqrt100)/sqrt500?

Oct 1, 2015

The answer is $\frac{7 \sqrt{5}}{5}$.

#### Explanation:

$\frac{7 \sqrt{100}}{\sqrt{500}}$

Numerator
Write the prime factorization for $100$ and simplify. Use the square root rule $\sqrt{{a}^{2}} = \left\mid a \right\mid$ .

$7 \sqrt{100} = 7 \sqrt{10 \times 10} = 7 \sqrt{{10}^{2}} = 7 \times 10 = 70 =$

$\frac{70}{\sqrt{500}}$

Denominator
Write the prime factorization for $\sqrt{500}$ and simplify. Again use the square root rule $\sqrt{{a}^{2}} = \left\mid a \right\mid$ .

$\sqrt{500} = \sqrt{2 \times 2 \times 5 \times 5 \times 5} = \sqrt{{2}^{2} \times {5}^{2} \times 5} = 2 \times 5 \sqrt{5} = 10 \sqrt{5}$

Recombine the numerator and denominator.

$\frac{70}{10 \sqrt{5}}$

Rationalize the denominator.

$\frac{7}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} =$

$\frac{7 \sqrt{5}}{\sqrt{25}}$

Write the prime factors for $25$.

$\sqrt{25} = \sqrt{5 \times 5} =$

$\sqrt{25} = \sqrt{{5}^{2}}$

Apply the square root rule $\sqrt{{a}^{2}} = \left\mid a \right\mid$ and simplify.

$\frac{7 \sqrt{5}}{5}$