Solve a^2/b^2-b^2/a^2/frac{1{{b}^2}+1/{a^2}}

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

Evaluate

$$a^{2}-b^{2}$$

Short Solution Steps

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

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

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

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

Do the multiplications in a^{2}a^{2}-b^{2}b^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b^{2} and a^{2} is a^{2}b^{2}. Multiply \frac{1}{b^{2}} times \frac{a^{2}}{a^{2}}. Multiply \frac{1}{a^{2}} times \frac{b^{2}}{b^{2}}.

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

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

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

Divide \frac{a^{4}-b^{4}}{a^{2}b^{2}} by \frac{a^{2}+b^{2}}{a^{2}b^{2}} by multiplying \frac{a^{4}-b^{4}}{a^{2}b^{2}} by the reciprocal of \frac{a^{2}+b^{2}}{a^{2}b^{2}}.

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

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

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

Factor the expressions that are not already factored.

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

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

$$\left(a+b\right)\left(a-b\right)$$

Expand the expression.

$$a^{2}-b^{2}$$

Expand

$$a^{2}-b^{2}$$

Short Solution Steps

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

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

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

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

Do the multiplications in a^{2}a^{2}-b^{2}b^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b^{2} and a^{2} is a^{2}b^{2}. Multiply \frac{1}{b^{2}} times \frac{a^{2}}{a^{2}}. Multiply \frac{1}{a^{2}} times \frac{b^{2}}{b^{2}}.

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

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

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

Divide \frac{a^{4}-b^{4}}{a^{2}b^{2}} by \frac{a^{2}+b^{2}}{a^{2}b^{2}} by multiplying \frac{a^{4}-b^{4}}{a^{2}b^{2}} by the reciprocal of \frac{a^{2}+b^{2}}{a^{2}b^{2}}.

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

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

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

Factor the expressions that are not already factored.

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

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

$$\left(a+b\right)\left(a-b\right)$$

Expand the expression.

$$a^{2}-b^{2}$$