Basic Math Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $n$ .

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Step 3.1.1

Move the leading negative in $- \frac{1}{n}$ into the numerator.

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $n$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Add $1$ and $1$ .

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

Step 4.5

Rewrite $1$ as $1^{2}$ .

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

Step 4.6

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 $b = 1$ .

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

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Multiply $n$ by $1$ .

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

Step 5.2

Cancel the common factor of $n + 1$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Divide $n - 1$ by $1$ .

$n - 1$