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

Mathematics

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Answer

−13-\frac{1}{3}

Step-by-step solution

5 steps

Here's the solution:

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

    (2+73)(2−73)=(2+7)(2−7)3×3\left( \frac{2+\sqrt{7}}{3} \right) \left( \frac{2-\sqrt{7}}{3} \right) = \frac{(2+\sqrt{7})(2-\sqrt{7})}{3 \times 3}

  2. Multiply the numerators 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

  3. Multiply the denominators.

    3×3=93 \times 3 = 9

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

    −39\frac{-3}{9}

  5. Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

    −3÷39÷3=−13\frac{-3 \div 3}{9 \div 3} = \frac{-1}{3}

    The simplified expression is:

    −13-\frac{1}{3}
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