Algebra Examples

$\frac{8 a^{3} - \frac{1}{27}}{{(2 a - \frac{1}{3})}^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

To write $8 a^{3}$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$\frac{8 a^{3} \cdot \frac{27}{27} - \frac{1}{27}}{{(2 a - \frac{1}{3})}^{2}}$

Step 1.2

Combine $8 a^{3}$ and $\frac{27}{27}$ .

$\frac{\frac{8 a^{3} \cdot 27}{27} - \frac{1}{27}}{{(2 a - \frac{1}{3})}^{2}}$

Step 1.3

Combine the numerators over the common denominator.

$\frac{\frac{8 a^{3} \cdot 27 - 1}{27}}{{(2 a - \frac{1}{3})}^{2}}$

Step 1.4

Rewrite $\frac{8 a^{3} \cdot 27 - 1}{27}$ in a factored form.

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Step 1.4.1

Rewrite $8 a^{3} \cdot 27$ as ${(2 a \cdot 3)}^{3}$ .

$\frac{\frac{{(2 a \cdot 3)}^{3} - 1}{27}}{{(2 a - \frac{1}{3})}^{2}}$

Step 1.4.2

Rewrite $1$ as $1^{3}$ .

$\frac{\frac{{(2 a \cdot 3)}^{3} - 1^{3}}{27}}{{(2 a - \frac{1}{3})}^{2}}$

Step 1.4.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 2 a \cdot 3$ and $b = 1$ .

$\frac{\frac{(2 a \cdot 3 - 1) ({(2 a \cdot 3)}^{2} + 2 a \cdot 3 \cdot 1 + 1^{2})}{27}}{{(2 a - \frac{1}{3})}^{2}}$

Step 1.4.4

Simplify.

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Step 1.4.4.1

Multiply $3$ by $2$ .

$\frac{\frac{(6 a - 1) ({(2 a \cdot 3)}^{2} + 2 a \cdot 3 \cdot 1 + 1^{2})}{27}}{{(2 a - \frac{1}{3})}^{2}}$

Step 1.4.4.2

Multiply $3$ by $2$ .

$\frac{\frac{(6 a - 1) ({(6 a)}^{2} + 2 a \cdot 3 \cdot 1 + 1^{2})}{27}}{{(2 a - \frac{1}{3})}^{2}}$

Step 1.4.4.3

Apply the product rule to $6 a$ .

$\frac{\frac{(6 a - 1) (6^{2} a^{2} + 2 a \cdot 3 \cdot 1 + 1^{2})}{27}}{{(2 a - \frac{1}{3})}^{2}}$

Step 1.4.4.4

Raise $6$ to the power of $2$ .

$\frac{\frac{(6 a - 1) (36 a^{2} + 2 a \cdot 3 \cdot 1 + 1^{2})}{27}}{{(2 a - \frac{1}{3})}^{2}}$

Step 1.4.4.5

Multiply $3$ by $2$ .

$\frac{\frac{(6 a - 1) (36 a^{2} + 6 a \cdot 1 + 1^{2})}{27}}{{(2 a - \frac{1}{3})}^{2}}$

Step 1.4.4.6

Multiply $6$ by $1$ .

$\frac{\frac{(6 a - 1) (36 a^{2} + 6 a + 1^{2})}{27}}{{(2 a - \frac{1}{3})}^{2}}$

Step 1.4.4.7

One to any power is one.

$\frac{\frac{(6 a - 1) (36 a^{2} + 6 a + 1)}{27}}{{(2 a - \frac{1}{3})}^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

To write $2 a$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{(6 a - 1) (36 a^{2} + 6 a + 1)}{27}}{{(2 a \cdot \frac{3}{3} - \frac{1}{3})}^{2}}$

Step 2.2

Combine $2 a$ and $\frac{3}{3}$ .

$\frac{\frac{(6 a - 1) (36 a^{2} + 6 a + 1)}{27}}{{(\frac{2 a \cdot 3}{3} - \frac{1}{3})}^{2}}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{\frac{(6 a - 1) (36 a^{2} + 6 a + 1)}{27}}{{(\frac{2 a \cdot 3 - 1}{3})}^{2}}$

Step 2.4

Multiply $3$ by $2$ .

$\frac{\frac{(6 a - 1) (36 a^{2} + 6 a + 1)}{27}}{{(\frac{6 a - 1}{3})}^{2}}$

Step 2.5

Apply the product rule to $\frac{6 a - 1}{3}$ .

$\frac{\frac{(6 a - 1) (36 a^{2} + 6 a + 1)}{27}}{\frac{{(6 a - 1)}^{2}}{3^{2}}}$

Step 2.6

Raise $3$ to the power of $2$ .

$\frac{\frac{(6 a - 1) (36 a^{2} + 6 a + 1)}{27}}{\frac{{(6 a - 1)}^{2}}{9}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{(6 a - 1) (36 a^{2} + 6 a + 1)}{27} \cdot \frac{9}{{(6 a - 1)}^{2}}$

Step 4

Combine.

$\frac{(6 a - 1) (36 a^{2} + 6 a + 1) \cdot 9}{27 {(6 a - 1)}^{2}}$

Step 5

Cancel the common factor of $6 a - 1$ and ${(6 a - 1)}^{2}$ .

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Step 5.1

Factor $6 a - 1$ out of $(6 a - 1) (36 a^{2} + 6 a + 1) \cdot 9$ .

$\frac{(6 a - 1) ((36 a^{2} + 6 a + 1) \cdot 9)}{27 {(6 a - 1)}^{2}}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $6 a - 1$ out of $27 {(6 a - 1)}^{2}$ .

$\frac{(6 a - 1) ((36 a^{2} + 6 a + 1) \cdot 9)}{(6 a - 1) (27 (6 a - 1))}$

Step 5.2.2

Cancel the common factor.

$\frac{(6 a - 1) ((36 a^{2} + 6 a + 1) \cdot 9)}{(6 a - 1) (27 (6 a - 1))}$

Step 5.2.3

Rewrite the expression.

$\frac{(36 a^{2} + 6 a + 1) \cdot 9}{27 (6 a - 1)}$

Step 6

Cancel the common factor of $9$ and $27$ .

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Step 6.1

Factor $9$ out of $(36 a^{2} + 6 a + 1) \cdot 9$ .

$\frac{9 \cdot (36 a^{2} + 6 a + 1)}{27 (6 a - 1)}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $9$ out of $27 (6 a - 1)$ .

$\frac{9 \cdot (36 a^{2} + 6 a + 1)}{9 (3 (6 a - 1))}$

Step 6.2.2

Cancel the common factor.

$\frac{9 \cdot (36 a^{2} + 6 a + 1)}{9 (3 (6 a - 1))}$

Step 6.2.3

Rewrite the expression.

$\frac{36 a^{2} + 6 a + 1}{3 (6 a - 1)}$