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Perform the following operations involving radicals and exponents: Add -3 extsqrt(11) + 4 extsqrt(11), Multiply 7 extsqrt(6) imes (2/(7 extsqrt(2))), and Divide (1/2)4 ext by 1/8.

Mathematics

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Answers

3 parts
1

11\sqrt{11}

2

232\sqrt{3}

3

12\frac{1}{2}

Step-by-step solution

9 steps

Here are the solutions to the questions:

Part 1· 2 steps

Add: −311+411-3\sqrt{11} + 4\sqrt{11}

  1. Identify like terms.

    Both terms, −311-3\sqrt{11} and 4114\sqrt{11}, have the same radical part, 11\sqrt{11}. This means they are like terms and can be combined.

  2. Combine the coefficients.

    Add the numerical coefficients while keeping the radical part the same.

    −311+411=(−3+4)11-3\sqrt{11} + 4\sqrt{11} = (-3+4)\sqrt{11} (−3+4)11=111(-3+4)\sqrt{11} = 1\sqrt{11} 111=111\sqrt{11} = \sqrt{11}

    The simplified expression is 11\sqrt{11}.

    11\boxed{\sqrt{11}}

Part 2

232\sqrt{3}

Multiply: 76×2727\sqrt{6} \times \frac{2}{7\sqrt{2}}

  1. Rewrite the expression as a single fraction.

    76×272=76×2727\sqrt{6} \times \frac{2}{7\sqrt{2}} = \frac{7\sqrt{6} \times 2}{7\sqrt{2}}

  2. Simplify by canceling common factors.

    The 7 in the numerator and the 7 in the denominator cancel out.

    76×272=262\frac{\cancel{7}\sqrt{6} \times 2}{\cancel{7}\sqrt{2}} = \frac{2\sqrt{6}}{\sqrt{2}}

  3. Simplify the radical expression.

    Use the property ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}.

    262=262\frac{2\sqrt{6}}{\sqrt{2}} = 2\sqrt{\frac{6}{2}} 262=232\sqrt{\frac{6}{2}} = 2\sqrt{3}

    The simplified expression is 232\sqrt{3}.

    23\boxed{2\sqrt{3}}

Part 3

12\frac{1}{2}

Divide: (12)4÷18(\frac{1}{2})^4 \div \frac{1}{8}

  1. Calculate the value of (12)4(\frac{1}{2})^4.

    (12)4=1424=1×1×1×12×2×2×2=116\left(\frac{1}{2}\right)^4 = \frac{1^4}{2^4} = \frac{1 \times 1 \times 1 \times 1}{2 \times 2 \times 2 \times 2} = \frac{1}{16}

  2. Rewrite the division as multiplication by the reciprocal.

    Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 18\frac{1}{8} is 81\frac{8}{1} or 8.

    116÷18=116×81\frac{1}{16} \div \frac{1}{8} = \frac{1}{16} \times \frac{8}{1}

  3. Multiply the fractions.

    116×81=1×816×1=816\frac{1}{16} \times \frac{8}{1} = \frac{1 \times 8}{16 \times 1} = \frac{8}{16}

  4. Simplify the resulting fraction.

    Divide both the numerator and the denominator by their greatest common divisor, which is 8.

    816=8÷816÷8=12\frac{8}{16} = \frac{8 \div 8}{16 \div 8} = \frac{1}{2}

    The simplified result is 12\frac{1}{2}.

    12\boxed{\frac{1}{2}}
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