Algebra Examples

$\frac{12 a^{2} - 3}{2} \cdot {(2 a + 1)}^{- 2} \cdot {(\frac{6}{(2 a + 1)})}^{- 1}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $3$ out of $12 a^{2} - 3$ .

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Step 1.1.1

Factor $3$ out of $12 a^{2}$ .

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

Step 1.1.2

Factor $3$ out of $- 3$ .

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

Step 1.1.3

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3

Combine.

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

Step 4

Multiply $3$ by $1$ .

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

Step 5

Cancel the common factor of $2 a + 1$ and ${(2 a + 1)}^{2}$ .

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Step 5.1

Factor $2 a + 1$ out of $3 (2 a + 1) (2 a - 1)$ .

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

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

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 6

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{3 (2 a - 1)}{2 (2 a + 1)} \cdot \frac{2 a + 1}{6}$

Step 7

Cancel the common factor of $3$ .

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Step 7.1

Factor $3$ out of $6$ .

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

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

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Step 8.1

Factor $2 a + 1$ out of $2 (2 a + 1)$ .

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

$\frac{2 a - 1}{2} \cdot \frac{1}{2}$

Step 9

Multiply $\frac{2 a - 1}{2}$ by $\frac{1}{2}$ .

$\frac{2 a - 1}{2 \cdot 2}$

Step 10

Multiply $2$ by $2$ .

$\frac{2 a - 1}{4}$