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Radicals and Absolute Values - Problem 1

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Simplifying square roots using absolute values. So for this example what we have is a square root of a big string of numbers and variables and what we are trying to simplify it.

So what I’m going to do is do this all at once starting from the left going to the right. Square root of 75 we know this, it's broken down into 25 and 3 square root of 25 is 5' so we have a 5 and then we are left with a root 3. Square root of x³ we can break this up into x² and x the square root of x² is just x and we have a x in left in the inside to make the third one.

But what we have to be careful about is x is a variable, we have a square root, this is the little invisible 2 here and we know that only positive numbers can come out of square roots, so what we have to do to make this positive is include an absolute value on that x term. Even root with a odd power coming out is an absolute value.

Square root of y to the 8th, y to the square root of y to the 8th is just y of 4th times y of 4th so this is just y to the 4rth. And for this one we actually don’t need an absolute value because anything to an even power is going to be positive. So y to the 4th by default this positive we don’t need to do anything else with it. And then the square root of z to the 5th, z to the 5th is close to z to the fourth which is (z²)². So we have z² and one z left over and again by the same logic we didn’t need an absolute value on y to the fourth we don’t need one on z² because any number squared is going to be positive.

So for this particular example we only need the absolute value around the x but it's good practice to kind of go through it and make sure you think about your degrees as they are coming out and making sure if they are actually going to be positive.

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