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Math Coach Amy
Amy Ferguson Moncure

H Algebra 2/Trig · Unit 1

Simplify Radicals

Write a square root in simplest radical form by pulling out a perfect square.

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Key Ideas Easy Medium Hard

√(c × d) = √c × √d

√12 = 2√3

What to Remember

Pull out a perfect square

Write the number as (perfect square) × (what is left). Then split the square root.

The formula is

√(c × d) = √c × √d.

Leave the leftover under the radical

The leftover factor should have no perfect-square factor other than 1.

Whole numbers stay unstacked

Write 4√3, not a fraction, when the index is 2 and the leftover is an integer.

Level 1

Easy

One perfect-square factor. Integer leftover.

Example
Simplify. √12

The formula is

√(c × d) = √c × √d

12 = 4 × 3

√12 = √(4 × 3)

√12 = √4 × √3

√4 = 2

2√3

Answer: 2√3

1
Simplify. √8
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The formula is

√(c × d) = √c × √d

8 = 4 × 2

√8 = √(4 × 2)

√8 = √4 × √2

√4 = 2

2√2

Answer: 2√2

2
Simplify. √18
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The formula is

√(c × d) = √c × √d

18 = 9 × 2

√18 = √(9 × 2)

√18 = √9 × √2

√9 = 3

3√2

Answer: 3√2

3
Simplify. √32
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The formula is

√(c × d) = √c × √d

32 = 16 × 2

√32 = √(16 × 2)

√32 = √16 × √2

√16 = 4

4√2

Answer: 4√2

4
Simplify. √50
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The formula is

√(c × d) = √c × √d

50 = 25 × 2

√50 = √(25 × 2)

√50 = √25 × √2

√25 = 5

5√2

Answer: 5√2

5
Simplify. √27
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The formula is

√(c × d) = √c × √d

27 = 9 × 3

√27 = √(9 × 3)

√27 = √9 × √3

√9 = 3

3√3

Answer: 3√3

Level 2

Medium

Larger perfect squares.

Example
Simplify. √80

The formula is

√(c × d) = √c × √d

80 = 16 × 5

√80 = √(16 × 5)

√80 = √16 × √5

√16 = 4

4√5

Answer: 4√5

1
Simplify. √48
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The formula is

√(c × d) = √c × √d

48 = 16 × 3

√48 = √(16 × 3)

√48 = √16 × √3

√16 = 4

4√3

Answer: 4√3

2
Simplify. √75
Show Answer

The formula is

√(c × d) = √c × √d

75 = 25 × 3

√75 = √(25 × 3)

√75 = √25 × √3

√25 = 5

5√3

Answer: 5√3

3
Simplify. √98
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The formula is

√(c × d) = √c × √d

98 = 49 × 2

√98 = √(49 × 2)

√98 = √49 × √2

√49 = 7

7√2

Answer: 7√2

4
Simplify. √45
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The formula is

√(c × d) = √c × √d

45 = 9 × 5

√45 = √(9 × 5)

√45 = √9 × √5

√9 = 3

3√5

Answer: 3√5

5
Simplify. √108
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The formula is

√(c × d) = √c × √d

108 = 36 × 3

√108 = √(36 × 3)

√108 = √36 × √3

√36 = 6

6√3

Answer: 6√3

Level 3

Hard

Bigger numbers. Still one leftover factor.

Example
Simplify. √192

The formula is

√(c × d) = √c × √d

192 = 64 × 3

√192 = √(64 × 3)

√192 = √64 × √3

√64 = 8

8√3

Answer: 8√3

1
Simplify. √200
Show Answer

The formula is

√(c × d) = √c × √d

200 = 100 × 2

√200 = √(100 × 2)

√200 = √100 × √2

√100 = 10

10√2

Answer: 10√2

2
Simplify. √288
Show Answer

The formula is

√(c × d) = √c × √d

288 = 144 × 2

√288 = √(144 × 2)

√288 = √144 × √2

√144 = 12

12√2

Answer: 12√2

3
Simplify. √300
Show Answer

The formula is

√(c × d) = √c × √d

300 = 100 × 3

√300 = √(100 × 3)

√300 = √100 × √3

√100 = 10

10√3

Answer: 10√3

4
Simplify. √320
Show Answer

The formula is

√(c × d) = √c × √d

320 = 64 × 5

√320 = √(64 × 5)

√320 = √64 × √5

√64 = 8

8√5

Answer: 8√5

5
Simplify. √147
Show Answer

The formula is

√(c × d) = √c × √d

147 = 49 × 3

√147 = √(49 × 3)

√147 = √49 × √3

√49 = 7

7√3

Answer: 7√3

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