Solve (frac{a^{2}}{a+B}-frac{a^{3}}{a^{2}+2aB+B^2}):(a/a+B-a^2/a^2-B^2)

$$( \frac { a ^ { 2 } } { a + B } - \frac { a ^ { 3 } } { a ^ { 2 } + 2 a B + B ^ { 2 } } ) : ( \frac { a } { a + B } - \frac { a ^ { 2 } } { a ^ { 2 } - B ^ { 2 } } )$$

Evaluate

$$-\frac{a\left(a-B\right)}{B+a}$$

Short Solution Steps

Factor a^{2}+2aB+B^{2}.

$$\frac{\frac{a^{2}}{a+B}-\frac{a^{3}}{\left(B+a\right)^{2}}}{\frac{a}{a+B}-\frac{a^{2}}{a^{2}-B^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+B and \left(B+a\right)^{2} is \left(B+a\right)^{2}. Multiply \frac{a^{2}}{a+B} times \frac{B+a}{B+a}.

$$\frac{\frac{a^{2}\left(B+a\right)}{\left(B+a\right)^{2}}-\frac{a^{3}}{\left(B+a\right)^{2}}}{\frac{a}{a+B}-\frac{a^{2}}{a^{2}-B^{2}}}$$

Since \frac{a^{2}\left(B+a\right)}{\left(B+a\right)^{2}} and \frac{a^{3}}{\left(B+a\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{a^{2}\left(B+a\right)-a^{3}}{\left(B+a\right)^{2}}}{\frac{a}{a+B}-\frac{a^{2}}{a^{2}-B^{2}}}$$

Do the multiplications in a^{2}\left(B+a\right)-a^{3}.

$$\frac{\frac{a^{2}B+a^{3}-a^{3}}{\left(B+a\right)^{2}}}{\frac{a}{a+B}-\frac{a^{2}}{a^{2}-B^{2}}}$$

Combine like terms in a^{2}B+a^{3}-a^{3}.

$$\frac{\frac{a^{2}B}{\left(B+a\right)^{2}}}{\frac{a}{a+B}-\frac{a^{2}}{a^{2}-B^{2}}}$$

Factor a^{2}-B^{2}.

$$\frac{\frac{a^{2}B}{\left(B+a\right)^{2}}}{\frac{a}{a+B}-\frac{a^{2}}{\left(B+a\right)\left(-B+a\right)}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+B and \left(B+a\right)\left(-B+a\right) is \left(B+a\right)\left(-B+a\right). Multiply \frac{a}{a+B} times \frac{-B+a}{-B+a}.

$$\frac{\frac{a^{2}B}{\left(B+a\right)^{2}}}{\frac{a\left(-B+a\right)}{\left(B+a\right)\left(-B+a\right)}-\frac{a^{2}}{\left(B+a\right)\left(-B+a\right)}}$$

Since \frac{a\left(-B+a\right)}{\left(B+a\right)\left(-B+a\right)} and \frac{a^{2}}{\left(B+a\right)\left(-B+a\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{a^{2}B}{\left(B+a\right)^{2}}}{\frac{a\left(-B+a\right)-a^{2}}{\left(B+a\right)\left(-B+a\right)}}$$

Do the multiplications in a\left(-B+a\right)-a^{2}.

$$\frac{\frac{a^{2}B}{\left(B+a\right)^{2}}}{\frac{-aB+a^{2}-a^{2}}{\left(B+a\right)\left(-B+a\right)}}$$

Combine like terms in -aB+a^{2}-a^{2}.

$$\frac{\frac{a^{2}B}{\left(B+a\right)^{2}}}{\frac{-aB}{\left(B+a\right)\left(-B+a\right)}}$$

Divide \frac{a^{2}B}{\left(B+a\right)^{2}} by \frac{-aB}{\left(B+a\right)\left(-B+a\right)} by multiplying \frac{a^{2}B}{\left(B+a\right)^{2}} by the reciprocal of \frac{-aB}{\left(B+a\right)\left(-B+a\right)}.

$$\frac{a^{2}B\left(B+a\right)\left(-B+a\right)}{\left(B+a\right)^{2}\left(-1\right)aB}$$

Cancel out Ba\left(B+a\right) in both numerator and denominator.

$$\frac{a\left(-B+a\right)}{-\left(B+a\right)}$$

Use the distributive property to multiply a by -B+a.

$$\frac{-aB+a^{2}}{-\left(B+a\right)}$$

To find the opposite of B+a, find the opposite of each term.

$$\frac{-aB+a^{2}}{-B-a}$$

Expand

$$-\frac{a^{2}-Ba}{B+a}$$

Short Solution Steps

Factor a^{2}+2aB+B^{2}.

$$\frac{\frac{a^{2}}{a+B}-\frac{a^{3}}{\left(B+a\right)^{2}}}{\frac{a}{a+B}-\frac{a^{2}}{a^{2}-B^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+B and \left(B+a\right)^{2} is \left(B+a\right)^{2}. Multiply \frac{a^{2}}{a+B} times \frac{B+a}{B+a}.

$$\frac{\frac{a^{2}\left(B+a\right)}{\left(B+a\right)^{2}}-\frac{a^{3}}{\left(B+a\right)^{2}}}{\frac{a}{a+B}-\frac{a^{2}}{a^{2}-B^{2}}}$$

Since \frac{a^{2}\left(B+a\right)}{\left(B+a\right)^{2}} and \frac{a^{3}}{\left(B+a\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{a^{2}\left(B+a\right)-a^{3}}{\left(B+a\right)^{2}}}{\frac{a}{a+B}-\frac{a^{2}}{a^{2}-B^{2}}}$$

Do the multiplications in a^{2}\left(B+a\right)-a^{3}.

$$\frac{\frac{a^{2}B+a^{3}-a^{3}}{\left(B+a\right)^{2}}}{\frac{a}{a+B}-\frac{a^{2}}{a^{2}-B^{2}}}$$

Combine like terms in a^{2}B+a^{3}-a^{3}.

$$\frac{\frac{a^{2}B}{\left(B+a\right)^{2}}}{\frac{a}{a+B}-\frac{a^{2}}{a^{2}-B^{2}}}$$

Factor a^{2}-B^{2}.

$$\frac{\frac{a^{2}B}{\left(B+a\right)^{2}}}{\frac{a}{a+B}-\frac{a^{2}}{\left(B+a\right)\left(-B+a\right)}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+B and \left(B+a\right)\left(-B+a\right) is \left(B+a\right)\left(-B+a\right). Multiply \frac{a}{a+B} times \frac{-B+a}{-B+a}.

$$\frac{\frac{a^{2}B}{\left(B+a\right)^{2}}}{\frac{a\left(-B+a\right)}{\left(B+a\right)\left(-B+a\right)}-\frac{a^{2}}{\left(B+a\right)\left(-B+a\right)}}$$

Since \frac{a\left(-B+a\right)}{\left(B+a\right)\left(-B+a\right)} and \frac{a^{2}}{\left(B+a\right)\left(-B+a\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{a^{2}B}{\left(B+a\right)^{2}}}{\frac{a\left(-B+a\right)-a^{2}}{\left(B+a\right)\left(-B+a\right)}}$$

Do the multiplications in a\left(-B+a\right)-a^{2}.

$$\frac{\frac{a^{2}B}{\left(B+a\right)^{2}}}{\frac{-aB+a^{2}-a^{2}}{\left(B+a\right)\left(-B+a\right)}}$$

Combine like terms in -aB+a^{2}-a^{2}.

$$\frac{\frac{a^{2}B}{\left(B+a\right)^{2}}}{\frac{-aB}{\left(B+a\right)\left(-B+a\right)}}$$

Divide \frac{a^{2}B}{\left(B+a\right)^{2}} by \frac{-aB}{\left(B+a\right)\left(-B+a\right)} by multiplying \frac{a^{2}B}{\left(B+a\right)^{2}} by the reciprocal of \frac{-aB}{\left(B+a\right)\left(-B+a\right)}.

$$\frac{a^{2}B\left(B+a\right)\left(-B+a\right)}{\left(B+a\right)^{2}\left(-1\right)aB}$$

Cancel out Ba\left(B+a\right) in both numerator and denominator.

$$\frac{a\left(-B+a\right)}{-\left(B+a\right)}$$

Use the distributive property to multiply a by -B+a.

$$\frac{-aB+a^{2}}{-\left(B+a\right)}$$

To find the opposite of B+a, find the opposite of each term.

$$\frac{-aB+a^{2}}{-B-a}$$