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Simplify the following radical expressions: i) sqrt(27) ii) sqrt(32) iii) sqrt(8)

Mathematics

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Answers

4 parts
i

333\sqrt{3}

ii

424\sqrt{2}

iii

222\sqrt{2}

iv

828\sqrt{2}

Step-by-step solution

16 steps

Here are the simplifications for each expression:

Part i· 4 steps

Simplify 27\sqrt{27}

  1. Find the largest perfect square factor of 27.

    The factors of 27 are 1, 3, 9, 27. The largest perfect square factor is 9.

  2. Rewrite the expression using the perfect square factor.

    27=9×3\sqrt{27} = \sqrt{9 \times 3}

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

    9×3=9×3\sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3}

  4. Simplify the perfect square root.

    9×3=33\sqrt{9} \times \sqrt{3} = 3\sqrt{3}

    The simplified expression is 33\boxed{3\sqrt{3}}.

Part ii

424\sqrt{2}

Simplify 32\sqrt{32}

  1. Find the largest perfect square factor of 32.

    The factors of 32 are 1, 2, 4, 8, 16, 32. The largest perfect square factor is 16.

  2. Rewrite the expression using the perfect square factor.

    32=16×2\sqrt{32} = \sqrt{16 \times 2}

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

    16×2=16×2\sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2}

  4. Simplify the perfect square root.

    16×2=42\sqrt{16} \times \sqrt{2} = 4\sqrt{2}

    The simplified expression is 42\boxed{4\sqrt{2}}.

Part iii

222\sqrt{2}

Simplify 8\sqrt{8}

  1. Find the largest perfect square factor of 8.

    The factors of 8 are 1, 2, 4, 8. The largest perfect square factor is 4.

  2. Rewrite the expression using the perfect square factor.

    8=4×2\sqrt{8} = \sqrt{4 \times 2}

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

    4×2=4×2\sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2}

  4. Simplify the perfect square root.

    4×2=22\sqrt{4} \times \sqrt{2} = 2\sqrt{2}

    The simplified expression is 22\boxed{2\sqrt{2}}.

Part iv

828\sqrt{2}

Simplify 128\sqrt{128}

  1. Find the largest perfect square factor of 128.

    The factors of 128 include 1, 2, 4, 8, 16, 32, 64, 128. The largest perfect square factor is 64.

  2. Rewrite the expression using the perfect square factor.

    128=64×2\sqrt{128} = \sqrt{64 \times 2}

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

    64×2=64×2\sqrt{64 \times 2} = \sqrt{64} \times \sqrt{2}

  4. Simplify the perfect square root.

    64×2=82\sqrt{64} \times \sqrt{2} = 8\sqrt{2}

    The simplified expression is 82\boxed{8\sqrt{2}}.

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