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Simplify the following: (a) sqrt(262)/sqrt(150) (b) sqrt(242) sqrt(98) (c) (sqrt(2) + sqrt(5)) (3sqrt(5) - sqrt(2))

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Answers

5 parts
a

425\frac{\sqrt{42}}{5}

b
154
c

13+21013 + 2\sqrt{10}

d

−3-\sqrt{3}

e

666\sqrt{6}

Step-by-step solution

13 steps

Here are the simplifications for each expression:

Part a· 3 steps

We need to simplify 252150\frac{\sqrt{252}}{\sqrt{150}}.

  1. Simplify the square roots in the numerator and denominator.

    252=36×7=67\sqrt{252} = \sqrt{36 \times 7} = 6\sqrt{7} 150=25×6=56\sqrt{150} = \sqrt{25 \times 6} = 5\sqrt{6}

  2. Substitute the simplified roots back into the fraction and rationalize the denominator.

    6756=6756×66=67×65×6=64230\frac{6\sqrt{7}}{5\sqrt{6}} = \frac{6\sqrt{7}}{5\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{6\sqrt{7 \times 6}}{5 \times 6} = \frac{6\sqrt{42}}{30}

  3. Simplify the fraction.

    64230=425\frac{6\sqrt{42}}{30} = \frac{\sqrt{42}}{5}

    The simplified expression is 425\boxed{\frac{\sqrt{42}}{5}}.

Part b154

We need to simplify 242×98\sqrt{242} \times \sqrt{98}.

  1. Simplify each square root.

    242=121×2=112\sqrt{242} = \sqrt{121 \times 2} = 11\sqrt{2} 98=49×2=72\sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2}

  2. Multiply the simplified square roots.

    (112)×(72)=11×7×2×2=77×2(11\sqrt{2}) \times (7\sqrt{2}) = 11 \times 7 \times \sqrt{2} \times \sqrt{2} = 77 \times 2

  3. Perform the multiplication.

    77×2=15477 \times 2 = 154

    The simplified expression is 154\boxed{154}.

Part c

13+21013 + 2\sqrt{10}

We need to simplify (2+5)(35−2)(\sqrt{2} + \sqrt{5})(3\sqrt{5} - \sqrt{2}).

  1. Expand the expression using the FOIL method (First, Outer, Inner, Last).

    (2+5)(35−2)=(2×35)+(2×−2)+(5×35)+(5×−2)(\sqrt{2} + \sqrt{5})(3\sqrt{5} - \sqrt{2}) = (\sqrt{2} \times 3\sqrt{5}) + (\sqrt{2} \times -\sqrt{2}) + (\sqrt{5} \times 3\sqrt{5}) + (\sqrt{5} \times -\sqrt{2}) =310−2+3×5−10= 3\sqrt{10} - 2 + 3 \times 5 - \sqrt{10} =310−2+15−10= 3\sqrt{10} - 2 + 15 - \sqrt{10}

  2. Combine like terms.

    (310−10)+(−2+15)=210+13(3\sqrt{10} - \sqrt{10}) + (-2 + 15) = 2\sqrt{10} + 13

    The simplified expression is 13+210\boxed{13 + 2\sqrt{10}}.

Part d

−3-\sqrt{3}

We need to simplify 12−27\sqrt{12} - \sqrt{27}.

  1. Simplify each square root.

    12=4×3=23\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} 27=9×3=33\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}

  2. Subtract the simplified square roots.

    23−33=(2−3)32\sqrt{3} - 3\sqrt{3} = (2 - 3)\sqrt{3}

  3. Perform the subtraction.

    (2−3)3=−3(2 - 3)\sqrt{3} = -\sqrt{3}

    The simplified expression is −3\boxed{-\sqrt{3}}.

Part e

666\sqrt{6}

We need to simplify (3×2)×(18×2)(\sqrt{3} \times \sqrt{2}) \times (\sqrt{18} \times \sqrt{2}).

  1. Simplify the terms inside each parenthesis.

    3×2=3×2=6\sqrt{3} \times \sqrt{2} = \sqrt{3 \times 2} = \sqrt{6} 18×2=18×2=36=6\sqrt{18} \times \sqrt{2} = \sqrt{18 \times 2} = \sqrt{36} = 6

  2. Multiply the simplified terms.

    6×6=66\sqrt{6} \times 6 = 6\sqrt{6}

    The simplified expression is 66\boxed{6\sqrt{6}}.

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