Simplify Square Roots and Cube Roots

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Key Ideas Easy Medium Hard Super Challenge

What to Remember

Pull out a perfect square

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

The square-root formula

√(c × d) = √c × √d.

Cube roots

Pull out a perfect cube. ∛(c × d) = ∛c × ∛d. Cube roots of negatives are negative.

Leftover has no perfect factor

The leftover under a square root has no perfect-square factor other than 1. Under a cube root, no perfect-cube factor other than 1.

Level 1

Easy

One perfect-square or perfect-cube factor.

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

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

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

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

Simplify. ∛16

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The formula is

∛(c × d) = ∛c × ∛d

16 = 8 × 2

∛16 = ∛(8 × 2)

∛16 = ∛8 × ∛2

∛8 = 2

2∛2

Answer: 2∛2

Simplify. ∛54

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The formula is

∛(c × d) = ∛c × ∛d

54 = 27 × 2

∛54 = ∛(27 × 2)

∛54 = ∛27 × ∛2

∛27 = 3

3∛2

Answer: 3∛2

Level 2

Medium

Larger perfect squares, or a negative cube root.

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

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

Simplify. √75

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The formula is

√(c × d) = √c × √d

75 = 25 × 3

√75 = √(25 × 3)

√75 = √25 × √3

√25 = 5

5√3

Answer: 5√3

Simplify. ∛40

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The formula is

∛(c × d) = ∛c × ∛d

40 = 8 × 5

∛40 = ∛(8 × 5)

∛40 = ∛8 × ∛5

∛8 = 2

2∛5

Answer: 2∛5

Simplify. ∛−16

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−16 = −8 × 2

∛−16 = ∛−8 × ∛2

−2∛2

Answer: −2∛2

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

Level 3

Hard

Bigger numbers. Product under a radical.

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

Simplify. √200

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The formula is

√(c × d) = √c × √d

200 = 100 × 2

√200 = √(100 × 2)

√200 = √100 × √2

√100 = 10

10√2

Answer: 10√2

Simplify. ∛192

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The formula is

∛(c × d) = ∛c × ∛d

192 = 64 × 3

∛192 = ∛(64 × 3)

∛192 = ∛64 × ∛3

∛64 = 4

4∛3

Answer: 4∛3

Simplify. √36 × 3

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√108

108 = 36 × 3

6√3

Answer: 6√3

Simplify. √722

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√72 = 6√2

6√22 = 3√2

Answer: 3√2

Simplify. ∛8 ×−2

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∛−16

−2∛2

Answer: −2∛2

Super Challenge

Which is simpler, √50 or 5√2? Show they are equal.

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50 = 25 × 2

√50 = 5√2

5√2 is simplest because 2 has no perfect-square factor.

Answer: 5√2

A student writes ∛16 = 2∛8. Correct the work.

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16 = 8 × 2, so ∛16 = ∛8 × ∛2 = 2∛2.

They swapped the leftover and the perfect cube.

Answer: 2∛2

Show that √12 × √3 = 6 two ways: multiply inside first, and simplify each radical first.

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√12 × √3 = √36 = 6

√12 = 2√3, then 2√3 × √3 = 2 × 3 = 6

Answer: 6

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