Question

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

Answer 1

`(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 `sqrt(49) / sqrt(500)`

Breaking into prime factors:
Notice that `49 = 7^2`
and `500 = 2^2 * 5^3`

Therefore `sqrt(49) / sqrt(500) = sqrt(7^2) / sqrt(2^2 * 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))#

`= (7sqrt(5)) / (2*5*5)` = `(7sqrt(5)) / 50`