How do you find the square root of -324/728?

1 Answer
Aug 14, 2016

Answer:

#sqrt(-324/728) = (9sqrt(182))/(182)i ~= 0.66712i#

Explanation:

#sqrt(-324/728) = sqrt(324/728)i#

All natural numbers can be expressed as the product of prime numbers. In evaluating the roots of rational numbers it is often useful to decompose the number into its prime factors.

Notice that: #324 = 2^2xx3^4#
And that: #728 = 2^3xx7xx13#

Hence: #sqrt(-324/728) = sqrt(2^2xx3^4)/sqrt( 2^3xx7xx13)i#

Since any pair of prime factors may be taken out of the #sqrt#

#sqrt(-324/728) = (2xx3xx3)/(2xxsqrt(2xx7xx13))i#

#=(cancel2xx3xx3)/(cancel2xxsqrt(2xx7xx13))i#

#=(9sqrt(2xx7xx13))/(2xx7xx13)i#

#= (9sqrt(182))/(182) i~= 0.66712i#