Solve (a+5b/a^{2-5ab}-a-5b/a^2+5ab)*25b^2-a^2/5b^2

$$( \frac { a + 5 b } { a ^ { 2 } - 5 a b } - \frac { a - 5 b } { a ^ { 2 } + 5 a b } ) \cdot \frac { 25 b ^ { 2 } - a ^ { 2 } } { 5 b ^ { 2 } }$$

Evaluate

$$-\frac{4}{b}$$

Short Solution Steps

Factor a^{2}-5ab. Factor a^{2}+5ab.

$$\left(\frac{a+5b}{a\left(a-5b\right)}-\frac{a-5b}{a\left(a+5b\right)}\right)\times \frac{25b^{2}-a^{2}}{5b^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a-5b\right) and a\left(a+5b\right) is a\left(a-5b\right)\left(a+5b\right). Multiply \frac{a+5b}{a\left(a-5b\right)} times \frac{a+5b}{a+5b}. Multiply \frac{a-5b}{a\left(a+5b\right)} times \frac{a-5b}{a-5b}.

$$\left(\frac{\left(a+5b\right)\left(a+5b\right)}{a\left(a-5b\right)\left(a+5b\right)}-\frac{\left(a-5b\right)\left(a-5b\right)}{a\left(a-5b\right)\left(a+5b\right)}\right)\times \frac{25b^{2}-a^{2}}{5b^{2}}$$

Since \frac{\left(a+5b\right)\left(a+5b\right)}{a\left(a-5b\right)\left(a+5b\right)} and \frac{\left(a-5b\right)\left(a-5b\right)}{a\left(a-5b\right)\left(a+5b\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(a+5b\right)\left(a+5b\right)-\left(a-5b\right)\left(a-5b\right)}{a\left(a-5b\right)\left(a+5b\right)}\times \frac{25b^{2}-a^{2}}{5b^{2}}$$

Do the multiplications in \left(a+5b\right)\left(a+5b\right)-\left(a-5b\right)\left(a-5b\right).

$$\frac{a^{2}+5ab+5ab+25b^{2}-a^{2}+5ab+5ab-25b^{2}}{a\left(a-5b\right)\left(a+5b\right)}\times \frac{25b^{2}-a^{2}}{5b^{2}}$$

Combine like terms in a^{2}+5ab+5ab+25b^{2}-a^{2}+5ab+5ab-25b^{2}.

$$\frac{20ab}{a\left(a-5b\right)\left(a+5b\right)}\times \frac{25b^{2}-a^{2}}{5b^{2}}$$

Cancel out a in both numerator and denominator.

$$\frac{20b}{\left(a-5b\right)\left(a+5b\right)}\times \frac{25b^{2}-a^{2}}{5b^{2}}$$

Multiply \frac{20b}{\left(a-5b\right)\left(a+5b\right)} times \frac{25b^{2}-a^{2}}{5b^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{20b\left(25b^{2}-a^{2}\right)}{\left(a-5b\right)\left(a+5b\right)\times 5b^{2}}$$

Cancel out 5b in both numerator and denominator.

$$\frac{4\left(-a^{2}+25b^{2}\right)}{b\left(a-5b\right)\left(a+5b\right)}$$

Factor the expressions that are not already factored.

$$\frac{4\left(a-5b\right)\left(-a-5b\right)}{b\left(a-5b\right)\left(a+5b\right)}$$

Extract the negative sign in -a-5b.

$$\frac{-4\left(a-5b\right)\left(a+5b\right)}{b\left(a-5b\right)\left(a+5b\right)}$$

Cancel out \left(a-5b\right)\left(a+5b\right) in both numerator and denominator.

$$\frac{-4}{b}$$

Expand

$$-\frac{4}{b}$$

Short Solution Steps

Factor a^{2}-5ab. Factor a^{2}+5ab.

$$\left(\frac{a+5b}{a\left(a-5b\right)}-\frac{a-5b}{a\left(a+5b\right)}\right)\times \frac{25b^{2}-a^{2}}{5b^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a-5b\right) and a\left(a+5b\right) is a\left(a-5b\right)\left(a+5b\right). Multiply \frac{a+5b}{a\left(a-5b\right)} times \frac{a+5b}{a+5b}. Multiply \frac{a-5b}{a\left(a+5b\right)} times \frac{a-5b}{a-5b}.

$$\left(\frac{\left(a+5b\right)\left(a+5b\right)}{a\left(a-5b\right)\left(a+5b\right)}-\frac{\left(a-5b\right)\left(a-5b\right)}{a\left(a-5b\right)\left(a+5b\right)}\right)\times \frac{25b^{2}-a^{2}}{5b^{2}}$$

Since \frac{\left(a+5b\right)\left(a+5b\right)}{a\left(a-5b\right)\left(a+5b\right)} and \frac{\left(a-5b\right)\left(a-5b\right)}{a\left(a-5b\right)\left(a+5b\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(a+5b\right)\left(a+5b\right)-\left(a-5b\right)\left(a-5b\right)}{a\left(a-5b\right)\left(a+5b\right)}\times \frac{25b^{2}-a^{2}}{5b^{2}}$$

Do the multiplications in \left(a+5b\right)\left(a+5b\right)-\left(a-5b\right)\left(a-5b\right).

$$\frac{a^{2}+5ab+5ab+25b^{2}-a^{2}+5ab+5ab-25b^{2}}{a\left(a-5b\right)\left(a+5b\right)}\times \frac{25b^{2}-a^{2}}{5b^{2}}$$

Combine like terms in a^{2}+5ab+5ab+25b^{2}-a^{2}+5ab+5ab-25b^{2}.

$$\frac{20ab}{a\left(a-5b\right)\left(a+5b\right)}\times \frac{25b^{2}-a^{2}}{5b^{2}}$$

Cancel out a in both numerator and denominator.

$$\frac{20b}{\left(a-5b\right)\left(a+5b\right)}\times \frac{25b^{2}-a^{2}}{5b^{2}}$$

Multiply \frac{20b}{\left(a-5b\right)\left(a+5b\right)} times \frac{25b^{2}-a^{2}}{5b^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{20b\left(25b^{2}-a^{2}\right)}{\left(a-5b\right)\left(a+5b\right)\times 5b^{2}}$$

Cancel out 5b in both numerator and denominator.

$$\frac{4\left(-a^{2}+25b^{2}\right)}{b\left(a-5b\right)\left(a+5b\right)}$$

Factor the expressions that are not already factored.

$$\frac{4\left(a-5b\right)\left(-a-5b\right)}{b\left(a-5b\right)\left(a+5b\right)}$$

Extract the negative sign in -a-5b.

$$\frac{-4\left(a-5b\right)\left(a+5b\right)}{b\left(a-5b\right)\left(a+5b\right)}$$

Cancel out \left(a-5b\right)\left(a+5b\right) in both numerator and denominator.

$$\frac{-4}{b}$$