Solve 5(a^2+5a+6)/3(a^2-64)+53(a+3)/60(a+8)

$$\frac { 5 ( a ^ { 2 } + 5 a + 6 ) } { 3 ( a ^ { 2 } - 64 ) } + \frac { 53 ( a + 3 ) } { 60 ( a + 8 ) }$$

Evaluate

$$\frac{\left(153a-224\right)\left(a+3\right)}{60\left(a^{2}-64\right)}$$

Short Solution Steps

Factor 3\left(a^{2}-64\right).

$$\frac{5\left(a^{2}+5a+6\right)}{3\left(a-8\right)\left(a+8\right)}+\frac{53\left(a+3\right)}{60\left(a+8\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(a-8\right)\left(a+8\right) and 60\left(a+8\right) is 60\left(a-8\right)\left(a+8\right). Multiply \frac{5\left(a^{2}+5a+6\right)}{3\left(a-8\right)\left(a+8\right)} times \frac{20}{20}. Multiply \frac{53\left(a+3\right)}{60\left(a+8\right)} times \frac{a-8}{a-8}.

$$\frac{20\times 5\left(a^{2}+5a+6\right)}{60\left(a-8\right)\left(a+8\right)}+\frac{53\left(a+3\right)\left(a-8\right)}{60\left(a-8\right)\left(a+8\right)}$$

Since \frac{20\times 5\left(a^{2}+5a+6\right)}{60\left(a-8\right)\left(a+8\right)} and \frac{53\left(a+3\right)\left(a-8\right)}{60\left(a-8\right)\left(a+8\right)} have the same denominator, add them by adding their numerators.

$$\frac{20\times 5\left(a^{2}+5a+6\right)+53\left(a+3\right)\left(a-8\right)}{60\left(a-8\right)\left(a+8\right)}$$

Do the multiplications in 20\times 5\left(a^{2}+5a+6\right)+53\left(a+3\right)\left(a-8\right).

$$\frac{100a^{2}+500a+600+53a^{2}-424a+159a-1272}{60\left(a-8\right)\left(a+8\right)}$$

Combine like terms in 100a^{2}+500a+600+53a^{2}-424a+159a-1272.

$$\frac{153a^{2}+235a-672}{60\left(a-8\right)\left(a+8\right)}$$

Expand 60\left(a-8\right)\left(a+8\right).

$$\frac{153a^{2}+235a-672}{60a^{2}-3840}$$

Expand

$$\frac{153a^{2}+235a-672}{60\left(a^{2}-64\right)}$$

Short Solution Steps

Factor 3\left(a^{2}-64\right).

$$\frac{5\left(a^{2}+5a+6\right)}{3\left(a-8\right)\left(a+8\right)}+\frac{53\left(a+3\right)}{60\left(a+8\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(a-8\right)\left(a+8\right) and 60\left(a+8\right) is 60\left(a-8\right)\left(a+8\right). Multiply \frac{5\left(a^{2}+5a+6\right)}{3\left(a-8\right)\left(a+8\right)} times \frac{20}{20}. Multiply \frac{53\left(a+3\right)}{60\left(a+8\right)} times \frac{a-8}{a-8}.

$$\frac{20\times 5\left(a^{2}+5a+6\right)}{60\left(a-8\right)\left(a+8\right)}+\frac{53\left(a+3\right)\left(a-8\right)}{60\left(a-8\right)\left(a+8\right)}$$

Since \frac{20\times 5\left(a^{2}+5a+6\right)}{60\left(a-8\right)\left(a+8\right)} and \frac{53\left(a+3\right)\left(a-8\right)}{60\left(a-8\right)\left(a+8\right)} have the same denominator, add them by adding their numerators.

$$\frac{20\times 5\left(a^{2}+5a+6\right)+53\left(a+3\right)\left(a-8\right)}{60\left(a-8\right)\left(a+8\right)}$$

Do the multiplications in 20\times 5\left(a^{2}+5a+6\right)+53\left(a+3\right)\left(a-8\right).

$$\frac{100a^{2}+500a+600+53a^{2}-424a+159a-1272}{60\left(a-8\right)\left(a+8\right)}$$

Combine like terms in 100a^{2}+500a+600+53a^{2}-424a+159a-1272.

$$\frac{153a^{2}+235a-672}{60\left(a-8\right)\left(a+8\right)}$$

Expand 60\left(a-8\right)\left(a+8\right).

$$\frac{153a^{2}+235a-672}{60a^{2}-3840}$$