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# Simplifying Radicals Study Guide (page 3)

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Updated on Aug 24, 2011

### When the Radicand Contains a Fraction

The radicand cannot be a fraction. If you get rid of the denominator in the radicand, then you no longer have a fraction. This process is called rationalizing the denominator. Your strategy will be to make the denominator a perfect square. To do that, you multiply the denominator by itself.However, if you multiply the denominator of a fraction by a number, you must multiply the numerator of the fraction by the same number. Take a look at the following examples.

Example:

Make the denominator a perfect square.

Take out the square roots. One is a perfect square and so is 2 · 2.

Example:

Make the denominator a perfect square.

The number 1 is considered a factor of all numbers. If the numerator does not contain a perfect square, then 1 will be the perfect square and will be in the numerator. Take the square root of 1 in the numerator and 3 · 3 in the denominator. The product of 2 · 3 will give you 6 for the radicand.

Example:

Make the denominator a perfect square.

Take the square roots.

### When a Radical Is in the Denominator

When you have a radical in the denominator, the expression is not in simplest form. The expression contains a radical in the denominator. To get rid of the radical in the denominator, rationalize the denominator. In other words, make the denominator a perfect square. To do that, you need to multiply the denominator by itself.

 Example: Simplify. The number 9 is a perfect square.

 Example: Rationalize the denominator. Simplify. Take the square root of 4.

 Example: Rationalize the denominator. Simplify. You aren't finished yet because both radicands contain perfect squares. Take the square root of 4. Finished? Not quite. You can divide 2 into 2, or cancel the 2's. √3

#### Tip

 Example: 2√5 + 7√5 Add the numbers in front of the radicals. 9√5

 Example: 11√5 – 4√5 Subtract the numbers in front of the radicals. 7√5

 Example: 4√3 + 2√5 + 6√3 You can add only the radicals that are the same. 10√3 + 2√5

 Example: 5√8 + 6√8 Add the radicals. 11√8 But √8 contains a factor that is a perfect square, so you aren't finished because your answer is not in simplest form. 11√2 · 4 Take out the square root of 4. 2 · 11√2 Simplify. 22√2

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