Algebra Examples

$\frac{2 p}{4 p^{2} - 1} \div \frac{6 p^{3}}{6 p + 3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2 p}{4 p^{2} - 1} \cdot \frac{6 p + 3}{6 p^{3}}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $2 p$ .

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Step 2.1.1

Factor $2 p$ out of $6 p^{3}$ .

$\frac{2 p}{4 p^{2} - 1} \cdot \frac{6 p + 3}{2 p (3 p^{2})}$

Step 2.1.2

Cancel the common factor.

$\frac{2 p}{4 p^{2} - 1} \cdot \frac{6 p + 3}{2 p (3 p^{2})}$

Step 2.1.3

Rewrite the expression.

$\frac{1}{4 p^{2} - 1} \cdot \frac{6 p + 3}{3 p^{2}}$

Step 2.2

Multiply $\frac{1}{4 p^{2} - 1}$ by $\frac{6 p + 3}{3 p^{2}}$ .

$\frac{6 p + 3}{(4 p^{2} - 1) (3 p^{2})}$

Step 2.3

Cancel the common factor of $6 p + 3$ and $3$ .

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Step 2.3.1

Factor $3$ out of $6 p$ .

$\frac{3 (2 p) + 3}{(4 p^{2} - 1) (3 p^{2})}$

Step 2.3.2

Factor $3$ out of $3$ .

$\frac{3 (2 p) + 3 \cdot 1}{(4 p^{2} - 1) (3 p^{2})}$

Step 2.3.3

Factor $3$ out of $3 (2 p) + 3 (1)$ .

$\frac{3 (2 p + 1)}{(4 p^{2} - 1) (3 p^{2})}$

Step 2.3.4

Cancel the common factors.

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Step 2.3.4.1

Factor $3$ out of $(4 p^{2} - 1) (3 p^{2})$ .

$\frac{3 (2 p + 1)}{3 ((4 p^{2} - 1) (p^{2}))}$

Step 2.3.4.2

Cancel the common factor.

$\frac{3 (2 p + 1)}{3 ((4 p^{2} - 1) (p^{2}))}$

Step 2.3.4.3

Rewrite the expression.

$\frac{2 p + 1}{(4 p^{2} - 1) (p^{2})}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $4 p^{2}$ as ${(2 p)}^{2}$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $2 p + 1$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

$\frac{1}{(2 p - 1) p^{2}}$

Step 4.2

Reorder factors in $\frac{1}{(2 p - 1) p^{2}}$ .

$\frac{1}{p^{2} (2 p - 1)}$