Simplify Square Roots of Quotients

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  • 0:02 The Quotient Rule
  • 0:19 Solving with Quotient Rule
  • 1:24 Rationalize the Denominator
  • 4:08 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

The quotient rule can be used to simplify square roots of quotients. This lesson will define the quotient rule and show you how it is used to simplify square roots.

The Quotient Rule

The quotient rule for radicals says that the radical of a quotient is the quotient of the radicals, which means:


Solve Square Roots with the Quotient Rule

You can use the quotient rule to solve radical expressions, like this.



We can't take the square root of either of these numbers, but we can use the quotient rule to simplify the expression.


75 divided by 3 is 25, which we can take the square root of.


Let's try another one.



We can use the quotient rule to simply this expression.


x3 divided by x is x2.

We can take the square root of x2.


And x is the answer.

Rationalize the Denominator

There are occasions when you will simplify a fraction with a radical and still end up with a square root in the denominator. When this happens, there is one more step you will have to do to complete the problem, and that is rationalize the denominator.



The first step is to use the quotient rule to make one fraction under the square root symbol.


We can simplify this fraction because both numbers are divisible by 3. This reduces the fraction to 5/2. Since there is still a fraction under the radical symbol, we must rationalize the denominator to fully simplify the expression.

To do this, first return the expression to a division problem containing two square roots using the quotient rule.


Now you need to multiply the numerator and denominator of the fraction by a number that will eliminate the radical in the denominator of the fraction. Remember, multiplying both the numerator and denominator of a fraction by the same number is like multiplying by 1, so you're not really changing the fraction. For this example, the number to multiply by is the square root of 2.


When you do this, you get:


Since the square root of 4 is 2, we can simplify one more step, leaving the answer without a radical in the denominator.


Let's try one more example.



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