To simplify the expression, let's first expand the denominator.
(3 + √7)² = (3 + √7)(3 + √7)
= 3(3) + 3(√7) + √7(3) + √7(√7)
= 9 + 3√7 + 3√7 + 7
= 16 + 6√7
Now, let's substitute this into the expression:
√7/(3 + √7)² = √7/(16 + 6√7)
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is (16 - 6√7):
√7/(3 + √7)² = (√7/(16 + 6√7)) * ((16 - 6√7)/(16 - 6√7))
= (√7 * (16 - 6√7))/(256 - 252)
= (16√7 - 6√49)/(4)
= (16√7 - 6(7))/(4)
= (16√7 - 42)/(4)
= (8√7 - 21)/2
Therefore, √7/(3 + √7)² simplifies to (8√7 - 21)/2 in surd form with a rational denominator.
Simplify √7/(3+√7)² giving your answer in surd form with a rational denominator
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