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

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

Evaluate

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

Short Solution Steps

Factor a^{2}-b^{2}.

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

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

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

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

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

Do the multiplications in 2a-b-2\left(a-b\right).

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

Combine like terms in 2a-b-2a+2b.

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

To raise \frac{b}{\left(a+b\right)\left(a-b\right)} to a power, raise both numerator and denominator to the power and then divide.

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

Factor a^{2}-b^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(a+b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}.

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

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

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

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

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

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

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

To raise \frac{b^{2}}{\left(a+b\right)\left(a-b\right)} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Cancel out \left(\left(a+b\right)\left(a-b\right)\right)^{2} in both numerator and denominator.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{b^{2}}{b^{4}}$$

Rewrite b^{4} as b^{2}b^{2}. Cancel out b^{2} in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor a^{2}-b^{2}.

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

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

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

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

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

Do the multiplications in 2a-b-2\left(a-b\right).

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

Combine like terms in 2a-b-2a+2b.

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

To raise \frac{b}{\left(a+b\right)\left(a-b\right)} to a power, raise both numerator and denominator to the power and then divide.

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

Factor a^{2}-b^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(a+b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}.

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

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

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

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

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

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

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

To raise \frac{b^{2}}{\left(a+b\right)\left(a-b\right)} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Cancel out \left(\left(a+b\right)\left(a-b\right)\right)^{2} in both numerator and denominator.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{b^{2}}{b^{4}}$$

Rewrite b^{4} as b^{2}b^{2}. Cancel out b^{2} in both numerator and denominator.

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