Simplify (n/3-3/n)/(1+n/3)

$\frac{\frac{n}{3} - \frac{3}{n}}{1 + \frac{n}{3}}$

Multiply the numerator and denominator of the complex fraction by $3 n$ .

Multiply $\frac{\frac{n}{3} - \frac{3}{n}}{1 + \frac{n}{3}}$ by $\frac{3 n}{3 n}$ .

$\frac{3 n}{3 n} \cdot \frac{\frac{n}{3} - \frac{3}{n}}{1 + \frac{n}{3}}$

Combine.

$\frac{3 n (\frac{n}{3} - \frac{3}{n})}{3 n (1 + \frac{n}{3})}$

Apply the distributive property.

$\frac{3 n \frac{n}{3} + 3 n (- \frac{3}{n})}{3 n \cdot 1 + 3 n \frac{n}{3}}$

Simplify by cancelling.

Cancel the common factor of $3$ .

Factor $3$ out of $3 n$ .

$\frac{3 (n) \frac{n}{3} + 3 n (- \frac{3}{n})}{3 n \cdot 1 + 3 n \frac{n}{3}}$

Cancel the common factor.

$\frac{3 n \frac{n}{3} + 3 n (- \frac{3}{n})}{3 n \cdot 1 + 3 n \frac{n}{3}}$

Rewrite the expression.

$\frac{n \cdot n + 3 n (- \frac{3}{n})}{3 n \cdot 1 + 3 n \frac{n}{3}}$

Raise $n$ to the power of $1$ .

$\frac{n^{1} n + 3 n (- \frac{3}{n})}{3 n \cdot 1 + 3 n \frac{n}{3}}$

Raise $n$ to the power of $1$ .

$\frac{n^{1} n^{1} + 3 n (- \frac{3}{n})}{3 n \cdot 1 + 3 n \frac{n}{3}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{n^{1 + 1} + 3 n (- \frac{3}{n})}{3 n \cdot 1 + 3 n \frac{n}{3}}$

Add $1$ and $1$ .

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

Cancel the common factor of $n$ .

Move the leading negative in $- \frac{3}{n}$ into the numerator.

$\frac{n^{2} + 3 n \frac{- 3}{n}}{3 n \cdot 1 + 3 n \frac{n}{3}}$

Factor $n$ out of $3 n$ .

$\frac{n^{2} + n \cdot 3 \frac{- 3}{n}}{3 n \cdot 1 + 3 n \frac{n}{3}}$

Cancel the common factor.

$\frac{n^{2} + n \cdot 3 \frac{- 3}{n}}{3 n \cdot 1 + 3 n \frac{n}{3}}$

Rewrite the expression.

$\frac{n^{2} + 3 \cdot - 3}{3 n \cdot 1 + 3 n \frac{n}{3}}$

Multiply $3$ by $- 3$ .

$\frac{n^{2} - 9}{3 n \cdot 1 + 3 n \frac{n}{3}}$

Cancel the common factor of $3$ .

Factor $3$ out of $3 n$ .

$\frac{n^{2} - 9}{3 n \cdot 1 + 3 (n) \frac{n}{3}}$

Cancel the common factor.

$\frac{n^{2} - 9}{3 n \cdot 1 + 3 n \frac{n}{3}}$

Rewrite the expression.

$\frac{n^{2} - 9}{3 n \cdot 1 + n \cdot n}$

Raise $n$ to the power of $1$ .

$\frac{n^{2} - 9}{3 n \cdot 1 + n^{1} n}$

Raise $n$ to the power of $1$ .

$\frac{n^{2} - 9}{3 n \cdot 1 + n^{1} n^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{n^{2} - 9}{3 n \cdot 1 + n^{1 + 1}}$

Add $1$ and $1$ .

$\frac{n^{2} - 9}{3 n \cdot 1 + n^{2}}$

Simplify the numerator.

Rewrite $9$ as $3^{2}$ .

$\frac{n^{2} - 3^{2}}{3 n \cdot 1 + n^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = n$ and $b = 3$ .

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

Simplify the denominator.

Factor $n$ out of $3 n \cdot 1 + n^{2}$ .

Factor $n$ out of $3 n \cdot 1$ .

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

Factor $n$ out of $n^{2}$ .

$\frac{(n + 3) (n - 3)}{n (3 \cdot 1) + n \cdot n}$

Factor $n$ out of $n (3 \cdot 1) + n \cdot n$ .

$\frac{(n + 3) (n - 3)}{n (3 \cdot 1 + n)}$

Multiply $3$ by $1$ .

$\frac{(n + 3) (n - 3)}{n (3 + n)}$

Cancel the common factor of $n + 3$ and $3 + n$ .

Reorder terms.

$\frac{(n + 3) (n - 3)}{n (n + 3)}$

Cancel the common factor.

$\frac{(n + 3) (n - 3)}{n (n + 3)}$

Rewrite the expression.

$\frac{n - 3}{n}$

Simplify (n/3-3/n)/(1+n/3)

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