Solve a^-4-b^-4/a^-2+b^-2

$$\frac { a ^ { - 4 } - b ^ { - 4 } } { a ^ { - 2 } + b ^ { - 2 } }$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored.

$$\frac{\left(-b^{-4}a^{4}+1\right)a^{-4}}{\left(b^{-2}a^{2}+1\right)a^{-2}}$$

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

$$\frac{-b^{-4}a^{4}+1}{\left(b^{-2}a^{2}+1\right)a^{2}}$$

Expand the expression.

$$\frac{1-\left(\frac{1}{b}a\right)^{4}}{a^{2}+b^{-2}a^{4}}$$

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

$$\frac{1-\left(\frac{a}{b}\right)^{4}}{a^{2}+b^{-2}a^{4}}$$

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

$$\frac{1-\frac{a^{4}}{b^{4}}}{a^{2}+b^{-2}a^{4}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{b^{4}}{b^{4}}.

$$\frac{\frac{b^{4}}{b^{4}}-\frac{a^{4}}{b^{4}}}{a^{2}+b^{-2}a^{4}}$$

Since \frac{b^{4}}{b^{4}} and \frac{a^{4}}{b^{4}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{b^{4}-a^{4}}{b^{4}}}{a^{2}+b^{-2}a^{4}}$$

Express \frac{\frac{b^{4}-a^{4}}{b^{4}}}{a^{2}+b^{-2}a^{4}} as a single fraction.

$$\frac{b^{4}-a^{4}}{b^{4}\left(a^{2}+b^{-2}a^{4}\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}{b^{-2}a^{2}\left(a^{2}+b^{2}\right)b^{4}}$$

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

$$\frac{\left(a+b\right)\left(-a+b\right)}{b^{-2}a^{2}b^{4}}$$

Expand the expression.

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

Expand \left(ab\right)^{2}.

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored.

$$\frac{\left(-b^{-4}a^{4}+1\right)a^{-4}}{\left(b^{-2}a^{2}+1\right)a^{-2}}$$

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

$$\frac{-b^{-4}a^{4}+1}{\left(b^{-2}a^{2}+1\right)a^{2}}$$

Expand the expression.

$$\frac{1-\left(\frac{1}{b}a\right)^{4}}{a^{2}+b^{-2}a^{4}}$$

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

$$\frac{1-\left(\frac{a}{b}\right)^{4}}{a^{2}+b^{-2}a^{4}}$$

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

$$\frac{1-\frac{a^{4}}{b^{4}}}{a^{2}+b^{-2}a^{4}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{b^{4}}{b^{4}}.

$$\frac{\frac{b^{4}}{b^{4}}-\frac{a^{4}}{b^{4}}}{a^{2}+b^{-2}a^{4}}$$

Since \frac{b^{4}}{b^{4}} and \frac{a^{4}}{b^{4}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{b^{4}-a^{4}}{b^{4}}}{a^{2}+b^{-2}a^{4}}$$

Express \frac{\frac{b^{4}-a^{4}}{b^{4}}}{a^{2}+b^{-2}a^{4}} as a single fraction.

$$\frac{b^{4}-a^{4}}{b^{4}\left(a^{2}+b^{-2}a^{4}\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(a+b\right)\left(-a+b\right)\left(a^{2}+b^{2}\right)}{b^{-2}a^{2}\left(a^{2}+b^{2}\right)b^{4}}$$

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

$$\frac{\left(a+b\right)\left(-a+b\right)}{b^{-2}a^{2}b^{4}}$$

Expand the expression.

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

Expand \left(ab\right)^{2}.

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