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

1 Answer
Mar 14, 2016

Answer:

#(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#