Solve (1/a-1/b)(1/a+1/b)div(frac{a}{b}-frac{b}{a})

$$( \frac { 1 } { a } - \frac { 1 } { b } ) ( \frac { 1 } { a } + \frac { 1 } { b } ) \div ( \frac { a } { b } - \frac { b } { a } )$$

Evaluate

$$-\frac{1}{ab}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a and b is ab. Multiply \frac{1}{a} times \frac{b}{b}. Multiply \frac{1}{b} times \frac{a}{a}.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a and b is ab. Multiply \frac{1}{a} times \frac{b}{b}. Multiply \frac{1}{b} times \frac{a}{a}.

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b and a is ab. Multiply \frac{a}{b} times \frac{a}{a}. Multiply \frac{b}{a} times \frac{b}{b}.

$$\frac{\frac{\left(b-a\right)\left(b+a\right)}{abab}}{\frac{aa}{ab}-\frac{bb}{ab}}$$

Since \frac{aa}{ab} and \frac{bb}{ab} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(b-a\right)\left(b+a\right)}{abab}}{\frac{aa-bb}{ab}}$$

Do the multiplications in aa-bb.

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

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

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

Cancel out ab in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in -a+b.

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

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

$$\frac{-1}{ab}$$

Expand

$$-\frac{1}{ab}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a and b is ab. Multiply \frac{1}{a} times \frac{b}{b}. Multiply \frac{1}{b} times \frac{a}{a}.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a and b is ab. Multiply \frac{1}{a} times \frac{b}{b}. Multiply \frac{1}{b} times \frac{a}{a}.

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b and a is ab. Multiply \frac{a}{b} times \frac{a}{a}. Multiply \frac{b}{a} times \frac{b}{b}.

$$\frac{\frac{\left(b-a\right)\left(b+a\right)}{abab}}{\frac{aa}{ab}-\frac{bb}{ab}}$$

Since \frac{aa}{ab} and \frac{bb}{ab} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(b-a\right)\left(b+a\right)}{abab}}{\frac{aa-bb}{ab}}$$

Do the multiplications in aa-bb.

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

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

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

Cancel out ab in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in -a+b.

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

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

$$\frac{-1}{ab}$$