Solve (ab/a-b+a)(ab/a+b-a):a^2b^2/b^2-a^2

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

Evaluate

$$\left(\frac{a}{b}\right)^{2}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{a-b}{a-b}.

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

Since \frac{ab}{a-b} and \frac{a\left(a-b\right)}{a-b} have the same denominator, add them by adding their numerators.

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

Do the multiplications in ab+a\left(a-b\right).

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{a+b}{a+b}.

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

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

$$\frac{\frac{a^{2}}{a-b}\times \frac{ab-a\left(a+b\right)}{a+b}}{\frac{a^{2}b^{2}}{b^{2}-a^{2}}}$$

Do the multiplications in ab-a\left(a+b\right).

$$\frac{\frac{a^{2}}{a-b}\times \frac{ab-a^{2}-ab}{a+b}}{\frac{a^{2}b^{2}}{b^{2}-a^{2}}}$$

Combine like terms in ab-a^{2}-ab.

$$\frac{\frac{a^{2}}{a-b}\times \frac{-a^{2}}{a+b}}{\frac{a^{2}b^{2}}{b^{2}-a^{2}}}$$

Multiply \frac{a^{2}}{a-b} times \frac{-a^{2}}{a+b} by multiplying numerator times numerator and denominator times denominator.

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

Divide \frac{a^{2}\left(-1\right)a^{2}}{\left(a-b\right)\left(a+b\right)} by \frac{a^{2}b^{2}}{b^{2}-a^{2}} by multiplying \frac{a^{2}\left(-1\right)a^{2}}{\left(a-b\right)\left(a+b\right)} by the reciprocal of \frac{a^{2}b^{2}}{b^{2}-a^{2}}.

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

Cancel out a^{2} in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in -a-b.

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

Cancel out \left(a+b\right)\left(a-b\right) in both numerator and denominator.

$$\frac{-\left(-1\right)a^{2}}{b^{2}}$$

Expand the expression.

$$\frac{a^{2}}{b^{2}}$$

Expand

$$\left(\frac{a}{b}\right)^{2}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{a-b}{a-b}.

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

Since \frac{ab}{a-b} and \frac{a\left(a-b\right)}{a-b} have the same denominator, add them by adding their numerators.

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

Do the multiplications in ab+a\left(a-b\right).

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{a+b}{a+b}.

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

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

$$\frac{\frac{a^{2}}{a-b}\times \frac{ab-a\left(a+b\right)}{a+b}}{\frac{a^{2}b^{2}}{b^{2}-a^{2}}}$$

Do the multiplications in ab-a\left(a+b\right).

$$\frac{\frac{a^{2}}{a-b}\times \frac{ab-a^{2}-ab}{a+b}}{\frac{a^{2}b^{2}}{b^{2}-a^{2}}}$$

Combine like terms in ab-a^{2}-ab.

$$\frac{\frac{a^{2}}{a-b}\times \frac{-a^{2}}{a+b}}{\frac{a^{2}b^{2}}{b^{2}-a^{2}}}$$

Multiply \frac{a^{2}}{a-b} times \frac{-a^{2}}{a+b} by multiplying numerator times numerator and denominator times denominator.

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

Divide \frac{a^{2}\left(-1\right)a^{2}}{\left(a-b\right)\left(a+b\right)} by \frac{a^{2}b^{2}}{b^{2}-a^{2}} by multiplying \frac{a^{2}\left(-1\right)a^{2}}{\left(a-b\right)\left(a+b\right)} by the reciprocal of \frac{a^{2}b^{2}}{b^{2}-a^{2}}.

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

Cancel out a^{2} in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in -a-b.

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

Cancel out \left(a+b\right)\left(a-b\right) in both numerator and denominator.

$$\frac{-\left(-1\right)a^{2}}{b^{2}}$$

Expand the expression.

$$\frac{a^{2}}{b^{2}}$$