How do you rationalize the denominator and simplify sqrt49/sqrt500?

Mar 14, 2016

$\frac{7 \sqrt{5}}{50}$

Explanation:

Every positive integer can be expressed as a product of prime numbers. This is helpful in evaluating roots of integers.

in this example we are asked to simplify $\frac{\sqrt{49}}{\sqrt{500}}$

Breaking into prime factors:
Notice that $49 = {7}^{2}$
and $500 = {2}^{2} \cdot {5}^{3}$

Therefore $\frac{\sqrt{49}}{\sqrt{500}} = \frac{\sqrt{{7}^{2}}}{\sqrt{{2}^{2} \cdot {5}^{3}}}$
Since we are evaluating square roots all powers of 2 may be taken through the root sign. Thus:

sqrt(7^2) / sqrt(2^2 * 5^3) = 7/ (2 * 5 sqrt(5)

To rationalize the denominator, multiply top and bottom by $\sqrt{5} :$

 = (7sqrt(5)) / (2*5*sqrt(5)*sqrt(5))

$= \frac{7 \sqrt{5}}{2 \cdot 5 \cdot 5}$ = $\frac{7 \sqrt{5}}{50}$