Solve 1/a+5b-a^2-3a^2a+6/a^2-25b^2+1/2a-10b*46

$$\frac { 1 } { a + 5 b } - \frac { a ^ { 2 } - 3 a ^ { 2 } a + 6 } { a ^ { 2 } - 25 b ^ { 2 } } + \frac { 1 } { 2 a - 10 b } \cdot 46$$

Evaluate

$$\frac{3a^{3}-a^{2}+24a+110b-6}{a^{2}-25b^{2}}$$

Short Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{1}{a+5b}-\frac{a^{2}-3a^{3}+6}{a^{2}-25b^{2}}+\frac{1}{2a-10b}\times 46$$

Express \frac{1}{2a-10b}\times 46 as a single fraction.

$$\frac{1}{a+5b}-\frac{a^{2}-3a^{3}+6}{a^{2}-25b^{2}}+\frac{46}{2a-10b}$$

Factor a^{2}-25b^{2}.

$$\frac{1}{a+5b}-\frac{a^{2}-3a^{3}+6}{\left(a-5b\right)\left(a+5b\right)}+\frac{46}{2a-10b}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+5b and \left(a-5b\right)\left(a+5b\right) is \left(a-5b\right)\left(a+5b\right). Multiply \frac{1}{a+5b} times \frac{a-5b}{a-5b}.

$$\frac{a-5b}{\left(a-5b\right)\left(a+5b\right)}-\frac{a^{2}-3a^{3}+6}{\left(a-5b\right)\left(a+5b\right)}+\frac{46}{2a-10b}$$

Since \frac{a-5b}{\left(a-5b\right)\left(a+5b\right)} and \frac{a^{2}-3a^{3}+6}{\left(a-5b\right)\left(a+5b\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a-5b-\left(a^{2}-3a^{3}+6\right)}{\left(a-5b\right)\left(a+5b\right)}+\frac{46}{2a-10b}$$

Do the multiplications in a-5b-\left(a^{2}-3a^{3}+6\right).

$$\frac{a-5b-a^{2}+3a^{3}-6}{\left(a-5b\right)\left(a+5b\right)}+\frac{46}{2a-10b}$$

Factor 2a-10b.

$$\frac{a-5b-a^{2}+3a^{3}-6}{\left(a-5b\right)\left(a+5b\right)}+\frac{46}{2\left(a-5b\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-5b\right)\left(a+5b\right) and 2\left(a-5b\right) is 2\left(a-5b\right)\left(a+5b\right). Multiply \frac{a-5b-a^{2}+3a^{3}-6}{\left(a-5b\right)\left(a+5b\right)} times \frac{2}{2}. Multiply \frac{46}{2\left(a-5b\right)} times \frac{a+5b}{a+5b}.

$$\frac{2\left(a-5b-a^{2}+3a^{3}-6\right)}{2\left(a-5b\right)\left(a+5b\right)}+\frac{46\left(a+5b\right)}{2\left(a-5b\right)\left(a+5b\right)}$$

Since \frac{2\left(a-5b-a^{2}+3a^{3}-6\right)}{2\left(a-5b\right)\left(a+5b\right)} and \frac{46\left(a+5b\right)}{2\left(a-5b\right)\left(a+5b\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(a-5b-a^{2}+3a^{3}-6\right)+46\left(a+5b\right)}{2\left(a-5b\right)\left(a+5b\right)}$$

Do the multiplications in 2\left(a-5b-a^{2}+3a^{3}-6\right)+46\left(a+5b\right).

$$\frac{2a-10b-2a^{2}+6a^{3}-12+46a+230b}{2\left(a-5b\right)\left(a+5b\right)}$$

Combine like terms in 2a-10b-2a^{2}+6a^{3}-12+46a+230b.

$$\frac{48a+220b-2a^{2}+6a^{3}-12}{2\left(a-5b\right)\left(a+5b\right)}$$

Factor the expressions that are not already factored in \frac{48a+220b-2a^{2}+6a^{3}-12}{2\left(a-5b\right)\left(a+5b\right)}.

$$\frac{2\left(3a^{3}-a^{2}+24a+110b-6\right)}{2\left(a-5b\right)\left(a+5b\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{3a^{3}-a^{2}+24a+110b-6}{\left(a-5b\right)\left(a+5b\right)}$$

Expand \left(a-5b\right)\left(a+5b\right).

$$\frac{3a^{3}-a^{2}+24a+110b-6}{a^{2}-25b^{2}}$$

Expand

$$-\frac{3a^{3}-a^{2}+24a+110b-6}{25b^{2}-a^{2}}$$

Short Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{1}{a+5b}-\frac{a^{2}-3a^{3}+6}{a^{2}-25b^{2}}+\frac{1}{2a-10b}\times 46$$

Express \frac{1}{2a-10b}\times 46 as a single fraction.

$$\frac{1}{a+5b}-\frac{a^{2}-3a^{3}+6}{a^{2}-25b^{2}}+\frac{46}{2a-10b}$$

Factor a^{2}-25b^{2}.

$$\frac{1}{a+5b}-\frac{a^{2}-3a^{3}+6}{\left(a-5b\right)\left(a+5b\right)}+\frac{46}{2a-10b}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+5b and \left(a-5b\right)\left(a+5b\right) is \left(a-5b\right)\left(a+5b\right). Multiply \frac{1}{a+5b} times \frac{a-5b}{a-5b}.

$$\frac{a-5b}{\left(a-5b\right)\left(a+5b\right)}-\frac{a^{2}-3a^{3}+6}{\left(a-5b\right)\left(a+5b\right)}+\frac{46}{2a-10b}$$

Since \frac{a-5b}{\left(a-5b\right)\left(a+5b\right)} and \frac{a^{2}-3a^{3}+6}{\left(a-5b\right)\left(a+5b\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a-5b-\left(a^{2}-3a^{3}+6\right)}{\left(a-5b\right)\left(a+5b\right)}+\frac{46}{2a-10b}$$

Do the multiplications in a-5b-\left(a^{2}-3a^{3}+6\right).

$$\frac{a-5b-a^{2}+3a^{3}-6}{\left(a-5b\right)\left(a+5b\right)}+\frac{46}{2a-10b}$$

Factor 2a-10b.

$$\frac{a-5b-a^{2}+3a^{3}-6}{\left(a-5b\right)\left(a+5b\right)}+\frac{46}{2\left(a-5b\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-5b\right)\left(a+5b\right) and 2\left(a-5b\right) is 2\left(a-5b\right)\left(a+5b\right). Multiply \frac{a-5b-a^{2}+3a^{3}-6}{\left(a-5b\right)\left(a+5b\right)} times \frac{2}{2}. Multiply \frac{46}{2\left(a-5b\right)} times \frac{a+5b}{a+5b}.

$$\frac{2\left(a-5b-a^{2}+3a^{3}-6\right)}{2\left(a-5b\right)\left(a+5b\right)}+\frac{46\left(a+5b\right)}{2\left(a-5b\right)\left(a+5b\right)}$$

Since \frac{2\left(a-5b-a^{2}+3a^{3}-6\right)}{2\left(a-5b\right)\left(a+5b\right)} and \frac{46\left(a+5b\right)}{2\left(a-5b\right)\left(a+5b\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(a-5b-a^{2}+3a^{3}-6\right)+46\left(a+5b\right)}{2\left(a-5b\right)\left(a+5b\right)}$$

Do the multiplications in 2\left(a-5b-a^{2}+3a^{3}-6\right)+46\left(a+5b\right).

$$\frac{2a-10b-2a^{2}+6a^{3}-12+46a+230b}{2\left(a-5b\right)\left(a+5b\right)}$$

Combine like terms in 2a-10b-2a^{2}+6a^{3}-12+46a+230b.

$$\frac{48a+220b-2a^{2}+6a^{3}-12}{2\left(a-5b\right)\left(a+5b\right)}$$

Factor the expressions that are not already factored in \frac{48a+220b-2a^{2}+6a^{3}-12}{2\left(a-5b\right)\left(a+5b\right)}.

$$\frac{2\left(3a^{3}-a^{2}+24a+110b-6\right)}{2\left(a-5b\right)\left(a+5b\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{3a^{3}-a^{2}+24a+110b-6}{\left(a-5b\right)\left(a+5b\right)}$$

Expand \left(a-5b\right)\left(a+5b\right).

$$\frac{3a^{3}-a^{2}+24a+110b-6}{a^{2}-25b^{2}}$$