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Simplify the following square root expressions: a) sqrt(12) b) sqrt(72) c) sqrt(80)

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5 parts
a

a) 23\text{a) } 2\sqrt{3}

b

b) 62\text{b) } 6\sqrt{2}

c

c) 45\text{c) } 4\sqrt{5}

d

d) 55\text{d) } 5\sqrt{5}

e

e) 105\text{e) } 10\sqrt{5}

Step-by-step solution

20 steps

Here are the simplified forms of the square root expressions:

Part a· 4 steps

Simplify 12\sqrt{12}

  1. Find the largest perfect square factor of 12.

    The factors of 12 are 1, 2, 3, 4, 6, 12. The largest perfect square factor is 4.

  2. Rewrite the expression using the perfect square factor.

    12=4×3\sqrt{12} = \sqrt{4 \times 3}

  3. Apply the property ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b}.

    4×3=4×3\sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3}

  4. Simplify the perfect square root.

    4×3=23\sqrt{4} \times \sqrt{3} = 2\sqrt{3} a) 23\boxed{\text{a) } 2\sqrt{3}}

Part b

b) 62\text{b) } 6\sqrt{2}

Simplify 72\sqrt{72}

  1. Find the largest perfect square factor of 72.

    The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. The largest perfect square factor is 36.

  2. Rewrite the expression using the perfect square factor.

    72=36×2\sqrt{72} = \sqrt{36 \times 2}

  3. Apply the property ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b}.

    36×2=36×2\sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2}

  4. Simplify the perfect square root.

    36×2=62\sqrt{36} \times \sqrt{2} = 6\sqrt{2} b) 62\boxed{\text{b) } 6\sqrt{2}}

Part c

c) 45\text{c) } 4\sqrt{5}

Simplify 80\sqrt{80}

  1. Find the largest perfect square factor of 80.

    The factors of 80 are 1, 2, 4, 5, 8, 10, 16, 20, 40, 80. The largest perfect square factor is 16.

  2. Rewrite the expression using the perfect square factor.

    80=16×5\sqrt{80} = \sqrt{16 \times 5}

  3. Apply the property ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b}.

    16×5=16×5\sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5}

  4. Simplify the perfect square root.

    16×5=45\sqrt{16} \times \sqrt{5} = 4\sqrt{5} c) 45\boxed{\text{c) } 4\sqrt{5}}

Part d

d) 55\text{d) } 5\sqrt{5}

Simplify 125\sqrt{125}

  1. Find the largest perfect square factor of 125.

    The factors of 125 are 1, 5, 25, 125. The largest perfect square factor is 25.

  2. Rewrite the expression using the perfect square factor.

    125=25×5\sqrt{125} = \sqrt{25 \times 5}

  3. Apply the property ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b}.

    25×5=25×5\sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5}

  4. Simplify the perfect square root.

    25×5=55\sqrt{25} \times \sqrt{5} = 5\sqrt{5} d) 55\boxed{\text{d) } 5\sqrt{5}}

Part e

e) 105\text{e) } 10\sqrt{5}

Simplify 500\sqrt{500}

  1. Find the largest perfect square factor of 500.

    The factors of 500 are 1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, 500. The largest perfect square factor is 100.

  2. Rewrite the expression using the perfect square factor.

    500=100×5\sqrt{500} = \sqrt{100 \times 5}

  3. Apply the property ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b}.

    100×5=100×5\sqrt{100 \times 5} = \sqrt{100} \times \sqrt{5}

  4. Simplify the perfect square root.

    100×5=105\sqrt{100} \times \sqrt{5} = 10\sqrt{5} e) 105\boxed{\text{e) } 10\sqrt{5}}
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