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Simplify the following expression: (3/(2+sqrt(7))) * (2/(2-sqrt(7)))

Mathematics

ScanSolve AI Answer

Answer

-2

Step-by-step solution

5 steps

Here's the solution:

  1. Combine the two fractions by multiplying the numerators and the denominators.

    (32+7)(22−7)=3×2(2+7)(2−7)\left( \frac{3}{2+\sqrt{7}} \right) \left( \frac{2}{2-\sqrt{7}} \right) = \frac{3 \times 2}{(2+\sqrt{7})(2-\sqrt{7})}

  2. Multiply the numerators.

    3×2=63 \times 2 = 6

  3. Multiply the denominators using the difference of squares formula, (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2.

    Here, a=2a=2 and b=7b=\sqrt{7}.

    (2+7)(2−7)=22−(7)2(2+\sqrt{7})(2-\sqrt{7}) = 2^2 - (\sqrt{7})^2 =4−7= 4 - 7 =−3= -3

  4. Substitute the simplified numerator and denominator back into the fraction.

    6−3\frac{6}{-3}

  5. Perform the division.

    6−3=−2\frac{6}{-3} = -2

    The simplified expression is:

    −2-2
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