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

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

Evaluate

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

Short Solution Steps

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

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

Since \frac{\left(a-1\right)\left(a+1\right)}{a^{2}b^{2}} and \frac{\left(1+2a^{2}\right)b^{2}}{a^{2}b^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(a-1\right)\left(a+1\right)-\left(1+2a^{2}\right)b^{2}.

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

Combine like terms in a^{2}+a-a-1-b^{2}-2a^{2}b^{2}.

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

Since \frac{a^{2}-1-2a^{2}b^{2}-b^{2}}{a^{2}b^{2}} and \frac{b^{2}+1}{a^{2}b^{2}} have the same denominator, add them by adding their numerators.

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

Combine like terms in a^{2}-1-2a^{2}b^{2}-b^{2}+b^{2}+1.

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

Factor the expressions that are not already factored in \frac{a^{2}-2a^{2}b^{2}}{a^{2}b^{2}}.

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

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

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

Expand

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

Short Solution Steps

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

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

Since \frac{\left(a-1\right)\left(a+1\right)}{a^{2}b^{2}} and \frac{\left(1+2a^{2}\right)b^{2}}{a^{2}b^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(a-1\right)\left(a+1\right)-\left(1+2a^{2}\right)b^{2}.

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

Combine like terms in a^{2}+a-a-1-b^{2}-2a^{2}b^{2}.

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

Since \frac{a^{2}-1-2a^{2}b^{2}-b^{2}}{a^{2}b^{2}} and \frac{b^{2}+1}{a^{2}b^{2}} have the same denominator, add them by adding their numerators.

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

Combine like terms in a^{2}-1-2a^{2}b^{2}-b^{2}+b^{2}+1.

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

Factor the expressions that are not already factored in \frac{a^{2}-2a^{2}b^{2}}{a^{2}b^{2}}.

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

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

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