Solve a*b^-1-a^-1*b/a^-2-b^-2

$$\frac { a \cdot b ^ { - 1 } - a ^ { - 1 } \cdot b } { a ^ { - 2 } - b ^ { - 2 } }$$

Evaluate

$$-ab$$

Short Solution Steps

Factor the expressions that are not already factored.

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

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

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

Expand the expression.

$$\frac{-ab+\frac{1}{b}a^{3}}{1-\left(\frac{1}{b}a\right)^{2}}$$

Express \frac{1}{b}a^{3} as a single fraction.

$$\frac{-ab+\frac{a^{3}}{b}}{1-\left(\frac{1}{b}a\right)^{2}}$$

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

$$\frac{-\frac{abb}{b}+\frac{a^{3}}{b}}{1-\left(\frac{1}{b}a\right)^{2}}$$

Since -\frac{abb}{b} and \frac{a^{3}}{b} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-abb+a^{3}}{b}}{1-\left(\frac{1}{b}a\right)^{2}}$$

Do the multiplications in -abb+a^{3}.

$$\frac{\frac{-ab^{2}+a^{3}}{b}}{1-\left(\frac{1}{b}a\right)^{2}}$$

Express \frac{1}{b}a as a single fraction.

$$\frac{\frac{-ab^{2}+a^{3}}{b}}{1-\left(\frac{a}{b}\right)^{2}}$$

To raise \frac{a}{b} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{-ab^{2}+a^{3}}{b}}{1-\frac{a^{2}}{b^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{b^{2}}{b^{2}}.

$$\frac{\frac{-ab^{2}+a^{3}}{b}}{\frac{b^{2}}{b^{2}}-\frac{a^{2}}{b^{2}}}$$

Since \frac{b^{2}}{b^{2}} and \frac{a^{2}}{b^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{-ab^{2}+a^{3}}{b}}{\frac{b^{2}-a^{2}}{b^{2}}}$$

Divide \frac{-ab^{2}+a^{3}}{b} by \frac{b^{2}-a^{2}}{b^{2}} by multiplying \frac{-ab^{2}+a^{3}}{b} by the reciprocal of \frac{b^{2}-a^{2}}{b^{2}}.

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

Cancel out b in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in a-b.

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

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

$$-ab$$

Expand

$$-ab$$

Short Solution Steps

Factor the expressions that are not already factored.

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

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

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

Expand the expression.

$$\frac{-ab+\frac{1}{b}a^{3}}{1-\left(\frac{1}{b}a\right)^{2}}$$

Express \frac{1}{b}a^{3} as a single fraction.

$$\frac{-ab+\frac{a^{3}}{b}}{1-\left(\frac{1}{b}a\right)^{2}}$$

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

$$\frac{-\frac{abb}{b}+\frac{a^{3}}{b}}{1-\left(\frac{1}{b}a\right)^{2}}$$

Since -\frac{abb}{b} and \frac{a^{3}}{b} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-abb+a^{3}}{b}}{1-\left(\frac{1}{b}a\right)^{2}}$$

Do the multiplications in -abb+a^{3}.

$$\frac{\frac{-ab^{2}+a^{3}}{b}}{1-\left(\frac{1}{b}a\right)^{2}}$$

Express \frac{1}{b}a as a single fraction.

$$\frac{\frac{-ab^{2}+a^{3}}{b}}{1-\left(\frac{a}{b}\right)^{2}}$$

To raise \frac{a}{b} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{-ab^{2}+a^{3}}{b}}{1-\frac{a^{2}}{b^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{b^{2}}{b^{2}}.

$$\frac{\frac{-ab^{2}+a^{3}}{b}}{\frac{b^{2}}{b^{2}}-\frac{a^{2}}{b^{2}}}$$

Since \frac{b^{2}}{b^{2}} and \frac{a^{2}}{b^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{-ab^{2}+a^{3}}{b}}{\frac{b^{2}-a^{2}}{b^{2}}}$$

Divide \frac{-ab^{2}+a^{3}}{b} by \frac{b^{2}-a^{2}}{b^{2}} by multiplying \frac{-ab^{2}+a^{3}}{b} by the reciprocal of \frac{b^{2}-a^{2}}{b^{2}}.

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

Cancel out b in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in a-b.

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

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

$$-ab$$