Algebra Examples

$\frac{a^{4}}{2 a^{2} - 4 a^{4}} \cdot \frac{8 a^{3} + 1}{2 a + 4 a^{2}}$

Step 1

Factor $2 a^{2}$ out of $2 a^{2} - 4 a^{4}$ .

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Step 1.1

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

$\frac{a^{4}}{2 a^{2} (1) - 4 a^{4}} \cdot \frac{8 a^{3} + 1}{2 a + 4 a^{2}}$

Step 1.2

Factor $2 a^{2}$ out of $- 4 a^{4}$ .

$\frac{a^{4}}{2 a^{2} (1) + 2 a^{2} (- 2 a^{2})} \cdot \frac{8 a^{3} + 1}{2 a + 4 a^{2}}$

Step 1.3

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

$\frac{a^{4}}{2 a^{2} (1 - 2 a^{2})} \cdot \frac{8 a^{3} + 1}{2 a + 4 a^{2}}$

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the product rule to $2 a$ .

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

Step 2.4.2

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

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

Step 2.4.3

Multiply $2$ by $- 1$ .

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

Step 2.4.4

Multiply $- 2$ by $1$ .

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

Step 2.4.5

One to any power is one.

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

Step 3

Simplify terms.

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Step 3.1

Factor $2 a$ out of $2 a + 4 a^{2}$ .

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Step 3.1.1

Factor $2 a$ out of $2 a$ .

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

Step 3.1.2

Factor $2 a$ out of $4 a^{2}$ .

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

Step 3.1.3

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

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

Step 3.2

Combine.

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

Step 4

Multiply $a^{2}$ by $a$ by adding the exponents.

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Step 4.1

Move $a$ .

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

Step 4.2

Multiply $a$ by $a^{2}$ .

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

Add $1$ and $2$ .

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

Step 5

Cancel the common factor of $a^{4}$ and $a^{3}$ .

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Step 5.1

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

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

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

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 6

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

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Step 6.1

Reorder terms.

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Multiply $2$ by $2$ .

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