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Multiplication of Surds Simplify the following: Example 5: (1) √12×√8 (2) √45×√27 (3) √28×√8×√12 (4) √8×√45×√90 (5) √12×3√20×2√45 (6) (2√3)² (7) (2√3)²×(2√2)

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Here are the solutions to the multiplication of surds problems:

1) 12×8\sqrt{12} \times \sqrt{8}

  1. Simplify each surd.

    12=4×3=23\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}
    8=4×2=22\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}

  2. Multiply the simplified surds.

    23×22=(2×2)×(3×2)=43×2=462\sqrt{3} \times 2\sqrt{2} = (2 \times 2) \times (\sqrt{3} \times \sqrt{2}) = 4\sqrt{3 \times 2} = \boxed{4\sqrt{6}}

    2) 45×27\sqrt{45} \times \sqrt{27}

  3. Simplify each surd.

    45=9×5=35\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}
    27=9×3=33\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}

  4. Multiply the simplified surds.

    35×33=(3×3)×(5×3)=95×3=9153\sqrt{5} \times 3\sqrt{3} = (3 \times 3) \times (\sqrt{5} \times \sqrt{3}) = 9\sqrt{5 \times 3} = \boxed{9\sqrt{15}}

    3) 28×8×12\sqrt{28} \times \sqrt{8} \times \sqrt{12}

  5. Simplify each surd.

    28=4×7=27\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}
    8=4×2=22\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
    12=4×3=23\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}

  6. Multiply the simplified surds.

    27×22×23=(2×2×2)×(7×2×3)=87×2×3=8422\sqrt{7} \times 2\sqrt{2} \times 2\sqrt{3} = (2 \times 2 \times 2) \times (\sqrt{7} \times \sqrt{2} \times \sqrt{3}) = 8\sqrt{7 \times 2 \times 3} = \boxed{8\sqrt{42}}

    4) 8×45×90\sqrt{8} \times \sqrt{45} \times \sqrt{90}

  7. Simplify each surd.

    8=4×2=22\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
    45=9×5=35\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}
    90=9×10=310\sqrt{90} = \sqrt{9 \times 10} = 3\sqrt{10}

  8. Multiply the simplified surds.

    22×35×310=(2×3×3)×(2×5×10)2\sqrt{2} \times 3\sqrt{5} \times 3\sqrt{10} = (2 \times 3 \times 3) \times (\sqrt{2} \times \sqrt{5} \times \sqrt{10}) =18×2×5×10=18×100=18×10=180= 18 \times \sqrt{2 \times 5 \times 10} = 18 \times \sqrt{100} = 18 \times 10 = \boxed{180}

    5) 12×320×245\sqrt{12} \times 3\sqrt{20} \times 2\sqrt{45}

  9. Simplify each surd.

    12=4×3=23\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}
    320=34×5=3×25=653\sqrt{20} = 3\sqrt{4 \times 5} = 3 \times 2\sqrt{5} = 6\sqrt{5}
    245=29×5=2×35=652\sqrt{45} = 2\sqrt{9 \times 5} = 2 \times 3\sqrt{5} = 6\sqrt{5}

  10. Multiply the simplified terms.

    23×65×65=(2×6×6)×(3×5×5)2\sqrt{3} \times 6\sqrt{5} \times 6\sqrt{5} = (2 \times 6 \times 6) \times (\sqrt{3} \times \sqrt{5} \times \sqrt{5}) =72×(3×5)=72×53=3603= 72 \times (\sqrt{3} \times 5) = 72 \times 5\sqrt{3} = \boxed{360\sqrt{3}}

    6) (23)2(2\sqrt{3})^2

  11. Apply the exponent to both parts of the term.

    (23)2=22×(3)2(2\sqrt{3})^2 = 2^2 \times (\sqrt{3})^2

  12. Calculate the squares and multiply.

    4×3=124 \times 3 = \boxed{12}

    7) (23)2×(22)(2\sqrt{3})^2 \times (2\sqrt{2})

  13. Calculate (23)2(2\sqrt{3})^2.

    (23)2=22×(3)2=4×3=12(2\sqrt{3})^2 = 2^2 \times (\sqrt{3})^2 = 4 \times 3 = 12

  14. Multiply this result by 222\sqrt{2}.

    12×22=(12×2)2=24212 \times 2\sqrt{2} = (12 \times 2)\sqrt{2} = \boxed{24\sqrt{2}}
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