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

1 Answer
Alan N.
Mar 14, 2016

(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

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