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Rationalize the denominator of the expression (1)/(sqrt(7)+2) and leave the answer in surd form.

Mathematics

ScanSolve AI Answer

Answer

7−25\frac{\sqrt{7}-\sqrt{2}}{5}

Step-by-step solution

4 steps

To leave the answer in surd form, we need to rationalize the denominator.

  1. Write the given expression.

    17+2\frac{1}{\sqrt{7}+\sqrt{2}}
  2. Multiply the numerator and the denominator by the conjugate of the denominator.

    The conjugate of 7+2\sqrt{7}+\sqrt{2} is 7−2\sqrt{7}-\sqrt{2}.

    17+2×7−27−2\frac{1}{\sqrt{7}+\sqrt{2}} \times \frac{\sqrt{7}-\sqrt{2}}{\sqrt{7}-\sqrt{2}}
  3. Perform the multiplication.

    Use the difference of squares formula (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2 for the denominator.

    1×(7−2)(7+2)(7−2)=7−2(7)2−(2)2\frac{1 \times (\sqrt{7}-\sqrt{2})}{(\sqrt{7}+\sqrt{2})(\sqrt{7}-\sqrt{2})} = \frac{\sqrt{7}-\sqrt{2}}{(\sqrt{7})^2 - (\sqrt{2})^2}
  4. Simplify the denominator.

    7−27−2=7−25\frac{\sqrt{7}-\sqrt{2}}{7 - 2} = \frac{\sqrt{7}-\sqrt{2}}{5}

    The simplified expression in surd form is 7−25\frac{\sqrt{7}-\sqrt{2}}{5}.

    7−25\boxed{\frac{\sqrt{7}-\sqrt{2}}{5}}

    Just one more to go today! Send it over.

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