What is Square root of 24 minus square root of 54 plus square root of 96?

1 Answer
Oct 8, 2015

Answer:

#3sqrt(6)#

Explanation:

Your starting expression looks like this

#sqrt(24) - sqrt(54) + sqrt(96)#

To try and simplify this expression, write out each value you have under a square root as a product of its prime factors.

This will get you

#24 = 2^3 * 3 = 2^2 * 2 * 3#

#54 = 2 * 3^3 = 2 * 3^2 * 3 = 3^2 * 2 * 3#

#96 = 2^5 * 3 = 2^4 * 2 * 3#

Notice that each number can be written as the product between a perfect square and #6#. This means that you can write

#sqrt(24) = sqrt(2^2 * 6) = sqrt(2^2) * sqrt(6) = 2sqrt(6)#

#sqrt(54) = sqrt(3^2 * 6) = sqrt(3^2) * sqrt(6) = 3sqrt(6)#

#sqrt(96) = sqrt(2^4 * 6) = sqrt(2^4) * sqrt(6) = 2^2sqrt(6) = 4sqrt(6)#

The expression can thus be written as

#2sqrt(6) - 3sqrt(6) + 4sqrt(6)#

which is equal to

#sqrt(6) * (2 - 3 + 4) = color(green)(3sqrt(6))#