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Simplifying Radicals Study Guide (page 2)

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Updated on Aug 24, 2011

When the Radicand Contains a Factor That Is a Perfect Square

To determine if a radicand contains any factors that are perfect squares, factor the radicand completely. All the factors must be prime. A number is prime if its only factors are 1 and the number itself. A prime number cannot be factored any further.

For example, here's how you simplify √12. The number 12 can be factored into 2 · 6. This is not completely factored because 6 is not prime. The number 6 can be further factored 2 · 3. The number 12 completely factored is 2 · 2 · 3.

The radical √12 can be written as √2 · 2 · 3. This can be split up into √2 · 2 · √3. Since √2 · 2 = 2, the simplified form of √12 is 2√3.

Example:18
Factor completely. 2 · 3 · 3
Separate out the perfect square 3 · 3. 3 · 3 · √2
Simplify. 3√2

Example:60

Factor completely.                                                                  

6 · 10

Neither 6 nor 10 is prime. Both can be factored further.

2 · 3 · 2 · 5

Separate out the perfect square                                     

2 · 2. √2 · 2 · √3 · 5

Because √3 · 5 contains no perfect squares, it cannot be simplified further.

2√15

Example:32
Factor completely. 2 · 16
The number 16 is not prime. It can be factored. 2 · 2 · 8
The number 8 is not prime. It can be factored. 2 · 2 · 2 · 4
The number 4 is not prime. It can be factored. 2 · 2 · 2 · 2 · 2

You have two sets of perfect squares, 2 · 2 and 2 · 2. The square root of each is 2, so you have two square roots of 2. The square roots go outside the radical. You then multiply the numbers that are outside the radical.

  2 · 2√2
Simplify. The product of 2 times 2 gives you 4. 4√2

 

Shortcut: You may have noticed in the first step, √2 · 16, that 16 is a perfect square, and the square root of 16 is 4. This would have given you the answer 4√2. Use the shorter method whenever you see one.

Example:50x3

Factor completely.              

2 · 5 · 5 · x · x · x

Separate the perfect square 5 · 5 and x · x.    

5 · 5 · √x · x · √2 · x

Simplify.    

5x2x

Example:9x2y3

Rewrite the radicand as the product of perfect squares.

9 x2 · y2 · y

Take out the square roots. 

3xyy

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