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Simplify the following: 1. sqrt(12) 2. sqrt(18)/sqrt(1) 3. cubert(18) * 3sqrt(7) 4. sqrt(26/4)

Mathematics

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Step-by-step solution

12 steps

Here are the solutions to the simplification problems:

  1. Simplify 12\sqrt{12}.
  1. Find the largest perfect square factor of 12.

    $$
    12 = 4 \times 3
    $$
    
  2. Apply the property ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b}.

    $$
    \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3}
    $$
    
  3. Simplify the perfect square root.

    $$
    \sqrt{4} = 2
    $$
    $$
    2 \times \sqrt{3} = 2\sqrt{3}
    $$
    The simplified expression is $\boxed{2\sqrt{3}}$.
    

    2. Simplify 189\frac{\sqrt{18}}{\sqrt{9}}.

  4. Simplify the denominator.

    $$
    \sqrt{9} = 3
    $$
    
  5. Rewrite the expression and simplify the numerator.

    $$
    \frac{\sqrt{18}}{3} = \frac{\sqrt{9 \times 2}}{3} = \frac{\sqrt{9} \times \sqrt{2}}{3}
    $$
    
  6. Simplify the perfect square root and cancel common factors.

    $$
    \frac{3\sqrt{2}}{3} = \sqrt{2}
    $$
    The simplified expression is $\boxed{\sqrt{2}}$.
    

    3. Simplify 373\sqrt[3]{\frac{3}{7}}.

  7. Apply the property abn=anbn\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}.

    $$
    \sqrt[3]{\frac{3}{7}} = \frac{\sqrt[3]{3}}{\sqrt[3]{7}}
    $$
    
  8. Rationalize the denominator by multiplying the numerator and denominator by 723\sqrt[3]{7^2}.

    $$
    \frac{\sqrt[3]{3}}{\sqrt[3]{7}} \times \frac{\sqrt[3]{7^2}}{\sqrt[3]{7^2}} = \frac{\sqrt[3]{3 \times 49}}{\sqrt[3]{7^3}}
    $$
    
  9. Simplify the denominator.

    $$
    \frac{\sqrt[3]{147}}{7}
    $$
    The simplified expression is $\boxed{\frac{\sqrt[3]{147}}{7}}$.
    

    4. Simplify 264\sqrt{\frac{26}{4}}.

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

    $$
    \sqrt{\frac{26}{4}} = \frac{\sqrt{26}}{\sqrt{4}}
    $$
    
  11. Simplify the denominator.

    $$
    \sqrt{4} = 2
    $$
    
  12. Rewrite the expression.

    $$
    \frac{\sqrt{26}}{2}
    $$
    The simplified expression is $\boxed{\frac{\sqrt{26}}{2}}$.
    
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