Basic Math Examples

$\frac{k^{2} - k}{k^{2} - 1} \div \frac{k + 1}{k^{2} + 2 k + 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Factor $k$ out of $k^{2} - k$ .

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Step 2.1

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

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

Step 2.2

Factor $k$ out of $- k$ .

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

Step 2.3

Factor $k$ out of $k \cdot k + k \cdot - 1$ .

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = k$ and $b = 1$ .

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

Step 4

Cancel the common factor of $k - 1$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

$\frac{k}{k + 1} \cdot \frac{k^{2} + 2 k + 1}{k + 1}$

Step 5

Factor using the perfect square rule.

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Step 5.1

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

$\frac{k}{k + 1} \cdot \frac{k^{2} + 2 k + 1^{2}}{k + 1}$

Step 5.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 k = 2 \cdot k \cdot 1$

Step 5.3

Rewrite the polynomial.

$\frac{k}{k + 1} \cdot \frac{k^{2} + 2 \cdot k \cdot 1 + 1^{2}}{k + 1}$

Step 5.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = k$ and $b = 1$ .

$\frac{k}{k + 1} \cdot \frac{{(k + 1)}^{2}}{k + 1}$

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $k + 1$ .

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Step 6.1.1

Factor $k + 1$ out of ${(k + 1)}^{2}$ .

$\frac{k}{k + 1} \cdot \frac{(k + 1) (k + 1)}{k + 1}$

Step 6.1.2

Cancel the common factor.

$\frac{k}{k + 1} \cdot \frac{(k + 1) (k + 1)}{k + 1}$

Step 6.1.3

Rewrite the expression.

$k \frac{k + 1}{k + 1}$

Step 6.2

Combine $k$ and $\frac{k + 1}{k + 1}$ .

$\frac{k (k + 1)}{k + 1}$

Step 6.3

Cancel the common factor of $k + 1$ .

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Step 6.3.1

Cancel the common factor.

$\frac{k (k + 1)}{k + 1}$

Step 6.3.2

Divide $k$ by $1$ .

$k$