Step-by-step explanation:
when we are asked to simplify an expression with fractions of square roots, it means we want to end up with an expression, where there are no square roots in the denominators (bottom part) of the remaining fractions. and for the numerators we try to do the usual simplification stuff.
also to remember : when transforming a fraction, we can always multiply by the value of 1, and it does not change its original value. that means we have to multiply denominator and numerator with the same thing (that way we multiply the fraction by 1).
sqrt(7)/(sqrt(2)-4) + 1/(2sqrt(7))
we can address both fractions first independently and see where this gets us.
I would start with the simpler 2nd fraction. how do we eliminate a square root ? by multiplying it by itself ... :
1/(2sqrt(7)) = 1/(2sqrt(7)) × sqrt(7)/sqrt(7) =
= sqrt(7)/(2sqrt(7)×sqrt(7)) =
= sqrt(7)/(2×7) = sqrt(7)/14
then, for the first fraction, we have to eliminate the square root of 2. but it is "caught" in a combination another term (- 4). what can we multiply the denominator with, to fully eliminate the square root ?
remember, (a + b)(a - b) = a² - b²
so, we are going to multiply by (sqrt(2) + 4) ... :
sqrt(7)/(sqrt(2) - 4) = sqrt(7)/(sqrt(2) - 4) × (sqrt(2) + 4)/(sqrt(2) + 4) =
= sqrt(7)×(sqrt(2) + 4) / ((sqrt(2) - 4)(sqrt(2) + 4)) =
= sqrt(7)×(sqrt(2) + 4) /
(sqrt(2)² - 4²) =
= sqrt(7)×(sqrt(2) + 4) /
(2 - 16) =
= sqrt(7)×(sqrt(2) + 4) / -14 =
= -sqrt(7)×(sqrt(2) + 4)/14
"miraculously" we have now 2 fractions with the same denominator (14). so, we can simply combine both numerators :
sqrt(7)/(sqrt(2)-4) + 1/(2sqrt(7)) =
= (-sqrt(7)×(sqrt(2) + 4) + sqrt(7))/14 =
= (-sqrt(7)×sqrt(2) - 4sqrt(7) + sqrt(7))/14 =
= (-sqrt(14) - 3sqrt(7))/14 =
= -(sqrt(14) + 3sqrt(7))/14
