Algebra Examples

$\frac{a^{4} - 81 b^{4}}{(9 b^{2} + a^{2}) (3 b - a)}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{{(a^{2})}^{2} - 81 b^{4}}{(9 b^{2} + a^{2}) (3 b - a)}$

Step 1.2

Rewrite $81 b^{4}$ as ${(9 b^{2})}^{2}$ .

$\frac{{(a^{2})}^{2} - {(9 b^{2})}^{2}}{(9 b^{2} + a^{2}) (3 b - a)}$

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a^{2}$ and $b = 9 b^{2}$ .

$\frac{(a^{2} + 9 b^{2}) (a^{2} - (9 b^{2}))}{(9 b^{2} + a^{2}) (3 b - a)}$

Step 1.4

Simplify.

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Step 1.4.1

Rewrite $9 b^{2}$ as ${(3 b)}^{2}$ .

$\frac{(a^{2} + 9 b^{2}) (a^{2} - {(3 b)}^{2})}{(9 b^{2} + a^{2}) (3 b - a)}$

Step 1.4.2

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

$\frac{(a^{2} + 9 b^{2}) ((a + 3 b) (a - (3 b)))}{(9 b^{2} + a^{2}) (3 b - a)}$

Step 1.4.3

Multiply $3$ by $- 1$ .

$\frac{(a^{2} + 9 b^{2}) (a + 3 b) (a - 3 b)}{(9 b^{2} + a^{2}) (3 b - a)}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $a^{2} + 9 b^{2}$ and $9 b^{2} + a^{2}$ .

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Step 2.1.1

Reorder terms.

$\frac{(a^{2} + 9 b^{2}) (a + 3 b) (a - 3 b)}{(a^{2} + 9 b^{2}) (3 b - a)}$

Step 2.1.2

Cancel the common factor.

$\frac{(a^{2} + 9 b^{2}) (a + 3 b) (a - 3 b)}{(a^{2} + 9 b^{2}) (3 b - a)}$

Step 2.1.3

Rewrite the expression.

$\frac{(a + 3 b) (a - 3 b)}{3 b - a}$

Step 2.2

Simplify terms.

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Step 2.2.1

Cancel the common factor of $a - 3 b$ and $3 b - a$ .

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Step 2.2.1.1

Factor $- 1$ out of $a$ .

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

Step 2.2.1.2

Factor $- 1$ out of $- 3 b$ .

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

Step 2.2.1.3

Factor $- 1$ out of $- 1 (- a) - (3 b)$ .

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

Step 2.2.1.4

Reorder terms.

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

Step 2.2.1.5

Cancel the common factor.

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

Step 2.2.1.6

Divide $(a + 3 b) \cdot (- 1)$ by $1$ .

$(a + 3 b) \cdot (- 1)$

Step 2.2.2

Apply the distributive property.

$a \cdot - 1 + 3 b \cdot - 1$

Step 2.2.3

Simplify the expression.

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Step 2.2.3.1

Move $- 1$ to the left of $a$ .

$- 1 \cdot a + 3 b \cdot - 1$

Step 2.2.3.2

Multiply $- 1$ by $3$ .

$- 1 \cdot a - 3 b$

Step 2.3

Rewrite $- 1 a$ as $- a$ .

$- a - 3 b$