Learning Objectives
Introduction
In this tutorial we will be looking at rewriting and simplifying radical expressions. Part of simplifying radicals is being able to take the root of an expression which is something that is shown in Tutorial 37: Radicals. It is good to be comfortable with radicals before entering College Algebra. I think you are ready to move ahead into the tutorial.
Tutorial
Note that if you have different index numbers, you CANNOT multiply them together.
Also, note that you can use this rule in either
direction depending on what your problem is asking you to do.
Since we cannot take the cube root of 6 and 6 does not have any factors
we can take the cube root of, this is as simplified as it gets.
Since we cannot take the fourth root of what's inside the radical sign
and 10 does not have any factors we can take the fourth root of, this is
as simplified as it gets.
A Quotient of Two Radicals
With the Same Index Number
If n is even, x and y represent
any nonnegative real number
and y does not equal 0.
If n is odd, x and y represent any real number and y does not equal 0.
This rule can also work in either direction.
*Square root of 25 is 5
*The cube root of 8 is 2
We can use the product rule of radicals in reverse to help us simplify the nth root of a number that we cannot take the nth root of as is, but has a factor that we can take the nth root of. If there is such a factor, we write the radicand as the product of that factor times the appropriate number and proceed.
We can also use the quotient rule to simplify a fraction that we have under the radical.
Note that the phrase "perfect square" means
that you can take the square root of it. Just as "perfect
cube" means we can take the cube root of the number, and so forth. I will be using that phrase in some of the following examples.
Check it out:
*Use the prod. rule of radicals to rewrite
*The square root of 100 is 10
Check it out:
*Use the prod. rule of radicals to rewrite
*The cube root of is 3ab
*Simplify fraction
50/5 simplifies to be 10. Since we cannot take the square root
of 10 and 10 does not have any factors we can take the square root of,
this is as simplified as it gets.
*Use the quotient rule of radicals to rewrite
*Simplify fraction
*Take the fourth root
Since is a
perfect fourth, we are able to take the fourth root of the whole radicand,
which leaves us with nothing under the radical sign.
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problem 1a: Use the product rule to multiply.
Practice Problem 2a: Use the quotient rule to simplify.
Practice Problems 3a - 3b: Simplify. Assume that the variables are positive.
Practice Problem 4a: Use the quotient rule to divide and the then simplify.
Assume that the variables are positive.
Need Extra Help on these Topics?
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Last revised on July 20, 2011 by Kim Seward.
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