Solve (a/a+2n-a+2n/2n)(a/a-2n-1+8n^3/8n^3-a^3)

$$( \frac { a } { a + 2 n } - \frac { a + 2 n } { 2 n } ) ( \frac { a } { a - 2 n } - 1 + \frac { 8 n ^ { 3 } } { 8 n ^ { 3 } - a ^ { 3 } } )$$

Evaluate

$$\frac{a}{2n-a}$$

Short Solution Steps

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

$$\left(\frac{a\times 2n}{2n\left(2n+a\right)}-\frac{\left(a+2n\right)\left(2n+a\right)}{2n\left(2n+a\right)}\right)\left(\frac{a}{a-2n}-1+\frac{8n^{3}}{8n^{3}-a^{3}}\right)$$

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

$$\frac{a\times 2n-\left(a+2n\right)\left(2n+a\right)}{2n\left(2n+a\right)}\left(\frac{a}{a-2n}-1+\frac{8n^{3}}{8n^{3}-a^{3}}\right)$$

Do the multiplications in a\times 2n-\left(a+2n\right)\left(2n+a\right).

$$\frac{a\times 2n-2an-a^{2}-4n^{2}-2an}{2n\left(2n+a\right)}\left(\frac{a}{a-2n}-1+\frac{8n^{3}}{8n^{3}-a^{3}}\right)$$

Combine like terms in a\times 2n-2an-a^{2}-4n^{2}-2an.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\left(\frac{a}{a-2n}-1+\frac{8n^{3}}{8n^{3}-a^{3}}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a-2n}{a-2n}.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\left(\frac{a}{a-2n}-\frac{a-2n}{a-2n}+\frac{8n^{3}}{8n^{3}-a^{3}}\right)$$

Since \frac{a}{a-2n} and \frac{a-2n}{a-2n} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\left(\frac{a-\left(a-2n\right)}{a-2n}+\frac{8n^{3}}{8n^{3}-a^{3}}\right)$$

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

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\left(\frac{a-a+2n}{a-2n}+\frac{8n^{3}}{8n^{3}-a^{3}}\right)$$

Combine like terms in a-a+2n.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\left(\frac{2n}{a-2n}+\frac{8n^{3}}{8n^{3}-a^{3}}\right)$$

Factor 8n^{3}-a^{3}.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\left(\frac{2n}{a-2n}+\frac{8n^{3}}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-2n and \left(2n-a\right)\left(4n^{2}+2an+a^{2}\right) is \left(2n-a\right)\left(4n^{2}+2an+a^{2}\right). Multiply \frac{2n}{a-2n} times \frac{-\left(4n^{2}+2an+a^{2}\right)}{-\left(4n^{2}+2an+a^{2}\right)}.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\left(\frac{2n\left(-1\right)\left(4n^{2}+2an+a^{2}\right)}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)}+\frac{8n^{3}}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)}\right)$$

Since \frac{2n\left(-1\right)\left(4n^{2}+2an+a^{2}\right)}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)} and \frac{8n^{3}}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)} have the same denominator, add them by adding their numerators.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\times \frac{2n\left(-1\right)\left(4n^{2}+2an+a^{2}\right)+8n^{3}}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)}$$

Do the multiplications in 2n\left(-1\right)\left(4n^{2}+2an+a^{2}\right)+8n^{3}.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\times \frac{-8n^{3}-4n^{2}a-2na^{2}+8n^{3}}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)}$$

Combine like terms in -8n^{3}-4n^{2}a-2na^{2}+8n^{3}.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\times \frac{-2na^{2}-4n^{2}a}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)}$$

Multiply \frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)} times \frac{-2na^{2}-4n^{2}a}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(-4n^{2}-2an-a^{2}\right)\left(-2na^{2}-4n^{2}a\right)}{2n\left(2n+a\right)\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)}$$

Extract the negative sign in -4n^{2}-2an-a^{2}.

$$\frac{-\left(4n^{2}+2an+a^{2}\right)\left(-4an^{2}-2na^{2}\right)}{2n\left(2n+a\right)\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)}$$

Cancel out 4n^{2}+2an+a^{2} in both numerator and denominator.

$$\frac{-\left(-4an^{2}-2na^{2}\right)}{2n\left(2n+a\right)\left(2n-a\right)}$$

Factor the expressions that are not already factored.

$$\frac{-2an\left(-2n-a\right)}{2n\left(2n+a\right)\left(2n-a\right)}$$

Extract the negative sign in -2n-a.

$$\frac{-\left(-1\right)\times 2an\left(2n+a\right)}{2n\left(2n+a\right)\left(2n-a\right)}$$

Cancel out 2n\left(2n+a\right) in both numerator and denominator.

$$\frac{-\left(-1\right)a}{2n-a}$$

Expand the expression.

$$\frac{a}{2n-a}$$

Expand

$$\frac{a}{2n-a}$$

Short Solution Steps

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

$$\left(\frac{a\times 2n}{2n\left(2n+a\right)}-\frac{\left(a+2n\right)\left(2n+a\right)}{2n\left(2n+a\right)}\right)\left(\frac{a}{a-2n}-1+\frac{8n^{3}}{8n^{3}-a^{3}}\right)$$

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

$$\frac{a\times 2n-\left(a+2n\right)\left(2n+a\right)}{2n\left(2n+a\right)}\left(\frac{a}{a-2n}-1+\frac{8n^{3}}{8n^{3}-a^{3}}\right)$$

Do the multiplications in a\times 2n-\left(a+2n\right)\left(2n+a\right).

$$\frac{a\times 2n-2an-a^{2}-4n^{2}-2an}{2n\left(2n+a\right)}\left(\frac{a}{a-2n}-1+\frac{8n^{3}}{8n^{3}-a^{3}}\right)$$

Combine like terms in a\times 2n-2an-a^{2}-4n^{2}-2an.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\left(\frac{a}{a-2n}-1+\frac{8n^{3}}{8n^{3}-a^{3}}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a-2n}{a-2n}.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\left(\frac{a}{a-2n}-\frac{a-2n}{a-2n}+\frac{8n^{3}}{8n^{3}-a^{3}}\right)$$

Since \frac{a}{a-2n} and \frac{a-2n}{a-2n} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\left(\frac{a-\left(a-2n\right)}{a-2n}+\frac{8n^{3}}{8n^{3}-a^{3}}\right)$$

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

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\left(\frac{a-a+2n}{a-2n}+\frac{8n^{3}}{8n^{3}-a^{3}}\right)$$

Combine like terms in a-a+2n.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\left(\frac{2n}{a-2n}+\frac{8n^{3}}{8n^{3}-a^{3}}\right)$$

Factor 8n^{3}-a^{3}.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\left(\frac{2n}{a-2n}+\frac{8n^{3}}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-2n and \left(2n-a\right)\left(4n^{2}+2an+a^{2}\right) is \left(2n-a\right)\left(4n^{2}+2an+a^{2}\right). Multiply \frac{2n}{a-2n} times \frac{-\left(4n^{2}+2an+a^{2}\right)}{-\left(4n^{2}+2an+a^{2}\right)}.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\left(\frac{2n\left(-1\right)\left(4n^{2}+2an+a^{2}\right)}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)}+\frac{8n^{3}}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)}\right)$$

Since \frac{2n\left(-1\right)\left(4n^{2}+2an+a^{2}\right)}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)} and \frac{8n^{3}}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)} have the same denominator, add them by adding their numerators.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\times \frac{2n\left(-1\right)\left(4n^{2}+2an+a^{2}\right)+8n^{3}}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)}$$

Do the multiplications in 2n\left(-1\right)\left(4n^{2}+2an+a^{2}\right)+8n^{3}.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\times \frac{-8n^{3}-4n^{2}a-2na^{2}+8n^{3}}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)}$$

Combine like terms in -8n^{3}-4n^{2}a-2na^{2}+8n^{3}.

$$\frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)}\times \frac{-2na^{2}-4n^{2}a}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)}$$

Multiply \frac{-4n^{2}-2an-a^{2}}{2n\left(2n+a\right)} times \frac{-2na^{2}-4n^{2}a}{\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(-4n^{2}-2an-a^{2}\right)\left(-2na^{2}-4n^{2}a\right)}{2n\left(2n+a\right)\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)}$$

Extract the negative sign in -4n^{2}-2an-a^{2}.

$$\frac{-\left(4n^{2}+2an+a^{2}\right)\left(-4an^{2}-2na^{2}\right)}{2n\left(2n+a\right)\left(2n-a\right)\left(4n^{2}+2an+a^{2}\right)}$$

Cancel out 4n^{2}+2an+a^{2} in both numerator and denominator.

$$\frac{-\left(-4an^{2}-2na^{2}\right)}{2n\left(2n+a\right)\left(2n-a\right)}$$

Factor the expressions that are not already factored.

$$\frac{-2an\left(-2n-a\right)}{2n\left(2n+a\right)\left(2n-a\right)}$$

Extract the negative sign in -2n-a.

$$\frac{-\left(-1\right)\times 2an\left(2n+a\right)}{2n\left(2n+a\right)\left(2n-a\right)}$$

Cancel out 2n\left(2n+a\right) in both numerator and denominator.

$$\frac{-\left(-1\right)a}{2n-a}$$

Expand the expression.

$$\frac{a}{2n-a}$$