Simplify √7/(3+√7)² giving your answer in surd form with a rational denominator
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.
To simplify the expression √7/(3+√7)², we need to rationalize the denominator.
First, let's expand the denominator: (3+√7)² = (3+√7)(3+√7).
Using the FOIL method, we get:
(3+√7)(3+√7) = 9 + 3√7 + 3√7 + √7√7
= 9 + 6√7 + 7
Simplifying further, we have:
9 + 6√7 + 7 = 16 + 6√7
Now, we rewrite the expression as:
√7 / (16 + 6√7)
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator, which is (16 - 6√7):
√7 / (16 + 6√7) * (16 - 6√7) / (16 - 6√7)
Multiplying the numerators and the denominators, we get:
(√7 * (16 - 6√7)) / ((16 + 6√7) * (16 - 6√7))
Expanding and simplifying the denominator:
= (√7 * 16 - √7 * 6√7) / (256 - (6√7)^2)
= (16√7 - 6√49) / (256 - 36 * 7)
= (16√7 - 6 * 7) / (256 - 252)
= (16√7 - 42) / 4
= (16√7 - 42)/4
Therefore, the simplified expression in surd form with a rational denominator is (16√7 - 42)/4.