How do you simplify square root of s^8 + square root of 25s^8 + 2 square root of s^8 - square root of s^4?

1 Answer
Sep 10, 2015

Answer:

#s^2 * (2sqrt(2)s + 1)(2sqrt(2)s-1)#

Explanation:

Your starting expression looks like this

#sqrt(s^8) + sqrt(25s^8) + 2sqrt(s^8) - sqrt(s^4)#

You can simplify this expression by playing around a bit with some properties of radicals and exponents. For example, you could say that

#sqrt(s^8) = sqrt((s^4)^2) = s^4#

Likewise,

#sqrt(s^4) = sqrt((s^2)^2) = s^2#

The second radical term can be written as

#sqrt(25 * s^8) = sqrt(25) * sqrt(s^8) = sqrt(5^2) * sqrt((s^4)^2) = 5 * s^4#

The expression will now take the form

#s^4 + 5 * s^4 + 2 * s^4 - s^2#

Group like-terms to get

#8s^4 - s^2#

You could simplify this further by using the difference of two squares factoring formula

#color(blue)(a^2 - b^2 = (a-b)(a+b))#

In your case, you could write

#8s^4 - s^2 = s^2 * (8s^2 - 1) = s^2 * [(sqrt(8) * s)^2 - 1^2]#

This can be simplified to

#s^2 * [(sqrt(8) * s)^2 - 1^2] = s^2 * (sqrt(8)s - 1) * (sqrt(8)s+1)#

The final form of the expression will be

#color(green)(s^2 * (2sqrt(2)s + 1)(2sqrt(2)s-1))#