Browse · harp

smc

algebra intermediate

Problem

The fraction

\frac{a ^{2} + x ^{2} - \frac{x ^{2} - a ^{2}}{a ^{2} + x ^{2}}}{a ^{2} + x ^{2}}

reduces to:

(A)

0

(B)

\frac{2 a ^{2}}{a ^{2} + x ^{2}}

(C)

\frac{2 x ^{2}}{( a ^{2} + x ^{2} ) ^{\frac{3}{2}}}

(D)

\frac{2 a ^{2}}{( a ^{2} + x ^{2} ) ^{\frac{3}{2}}}

Solution

Multiplying the numerator and denominator by

a^{2} + x^{2}

results in

\frac{a ^{2} + x ^{2} - x ^{2} + a ^{2}}{( a ^{2} + x ^{2} ) ( a ^{2} + x ^{2} )} = \frac{2 a ^{2}}{( a ^{2} + x ^{2} ) ( a ^{2} + x ^{2} )} .

Since

a^{2} + x^{2} = (a^{2} + x^{2})^{\frac{1}{2}}

, the denominator is

(a^{2} + x^{2})^{2} \cdot (a^{2} + x^{2})^{\frac{1}{2}} = (a^{2} + x^{2})^{\frac{3}{2}}

\frac{2 a ^{2}}{( a ^{2} + x ^{2} ) ^{\frac{3}{2}}}

.

Final answer

D