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
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))#
