To simplify the expression, rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of sqrt(7) + sqrt(2) is $sqrt(7) - sqrt(2)
This languages problem is solved step by step below, with detailed explanations to help you understand the method and arrive at the correct answer.
ScanSolve AI Answer
Step 1: To simplify the expression, rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of $\sqrt{7} + \sqrt{2}$ is $\sqrt{7} - \sqrt{2}$.
$$ \frac{1}{\sqrt{7} + \sqrt{2}} \times \frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} - \sqrt{2}} $$
Step 2: Multiply the numerators.
$$ 1 \times (\sqrt{7} - \sqrt{2}) = \sqrt{7} - \sqrt{2} $$
Step 3: Multiply the denominators using the difference of squares formula $(a+b)(a-b) = a^2 - b^2$.
$$ (\sqrt{7} + \sqrt{2})(\sqrt{7} - \sqrt{2}) = (\sqrt{7})^2 - (\sqrt{2})^2 $$
$$ = 7 - 2 $$
$$ = 5 $$
Step 4: Combine the simplified numerator and denominator.
$$ \frac{\sqrt{7} - \sqrt{2}}{5} $$
The simplified expression is:
$$ \boxed{\frac{\sqrt{7} - \sqrt{2}}{5}} $$
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