The fraction √a^2+x^2-x^2-a^2√a^2+x^2a^2+x^2 reduces to:

1953 AMC 12 Problem 7

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The fraction

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

reduces to:

0

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

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

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

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

Combine the two terms in the numerator over a common denominator

The new numerator simplifies to

(a^2+x^2)-(x^2-a^2)

The numerator is

\begin{aligned} &\frac{a^2+x^2-(x^2-a^2)} {\sqrt{a^2+x^2}}\\ &\qquad{}=\frac{2a^2}{\sqrt{a^2+x^2}}. \end{aligned}

Dividing by

a^2+x^2

gives

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

Thus, the correct answer is D.