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# Simplifying radical expressions (subtraction)

Sal simplifies 4∜(81x⁵)-2∜(81x⁵)-√(x³). Created by Sal Khan and Monterey Institute for Technology and Education.

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• What is the difference between a square root and a principle square root?
• Since both (-x)^2 and x^2 equal x^2, we use the phrase principal square root to note that we are only interested in the positive value. Most 'real world' applications of radical expressions are not concerned with the negative square root (negative length, or a negative amount of time, for example) so we ignore them by focusing on the principal quantity.
• At the very end, isn't it possible to factor out a |x|?
• Yes you can factor it out. See https://www.wolframalpha.com/input/?i=is+6*|x|*x^%281%2F4%29-|x|*sqrt%28x%29+equal+to+|x|*%286*x^%281%2F4%29-sqrt%28x%29%29
So the fully simplified expression is |x|*(6*x^(1/4)-x^(1/2))
• I tried to solve this myself before I finished the video to see how Sal solved the expression, and I got a differently simplified answer. I turned the root of 4 into a fractional exponent and turned the square root into a fractional exponent as well. I presume that it's still technically the same answer, but just either not simplified completely or a different way of simplifying.

This is what I did:
4*4√(81x^5) - 2*4√(81x^5) - √(x^3)
2*4√(81x^5) - √(x^3)
2*(81x^5)^1/4 - (x^3)^1/2
2*3*x^5/4 - x^3/2
6x^5/4 - x^3/2
So if this is just not completely simplified, I guess the real question is, how do you simplify it further from here? Would you have to go back and do it the way Sal do it or is it possible to continue simplifying using this method?
• Couldn't you get rid of the ∜x and √x by converting to x^1/4 and x^1/2, then putting them together as x^1/4 and x^2/4?... Or is this rulled out because it would result in a negative root?
• wouldnt 81 be squared into 9????? Or am i missing something?
• It doesn't ask for the square root, but the 4th.
• Why is 3 not an absolute value while x is?
• 3 is just a number, if you take the absolute value of 3, that's just 3. We need to take the absolute value of x because x is a variable, it can be positive or negative. 3 isn't a variable, it can't be positive or negative, it's always 3.
Although, as Sal noted, you might not need the absolute value signs anyway, since we will probably only be dealing with positive values of x in this particular expression, but it can't hurt to leave them in.
• How can I do this simplification/subtraction with fractions? For example say : Sqrt(6)/6 - Sqrt(2)/4?
• why couldn't he do 9x9 instead of 3^4 at the beginning
• Then you would not be able to take and cancel out the ^4. You would end up having to make it 3^4 anyways. So he just did it earlier.
(1 vote)
• I have followed another path and it seems to offer a further simplification, I want to know if this does not work...

1. Undistribute the 4th root expression convert to a fraction exponent
(4-2)(3x^5/4)-x^3/2
No absolute value is required from this because both exponents have an odd numerator which would resolve a negative x into a negative radicant and it would not therefore be possible to take a principal 4th root.

This can be further simplified by creating a common denominator between the two fractional exponents.
(6x^5/4)-(x^5/4 * x^1/4)
And factor out the common expression...
x^5/4(6-x^1/4)

I would like to know if this line of reasoning is correct, and if so why we would even need to consider the absolute value of x if we always keep the exponent odd inside a radical?