Pre-Algebra Examples

$\frac{9 x^{2} - 24 x + 64}{27 x^{3} + 512}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $27 x^{3}$ as ${(3 x)}^{3}$ .

$\frac{9 x^{2} - 24 x + 64}{{(3 x)}^{3} + 512}$

Step 1.2

Rewrite $512$ as $8^{3}$ .

$\frac{9 x^{2} - 24 x + 64}{{(3 x)}^{3} + 8^{3}}$

Step 1.3

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 $a = 3 x$ and $b = 8$ .

$\frac{9 x^{2} - 24 x + 64}{(3 x + 8) ({(3 x)}^{2} - (3 x) \cdot 8 + 8^{2})}$

Step 1.4

Simplify.

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Step 1.4.1

Apply the product rule to $3 x$ .

$\frac{9 x^{2} - 24 x + 64}{(3 x + 8) (3^{2} x^{2} - (3 x) \cdot 8 + 8^{2})}$

Step 1.4.2

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

$\frac{9 x^{2} - 24 x + 64}{(3 x + 8) (9 x^{2} - (3 x) \cdot 8 + 8^{2})}$

Step 1.4.3

Multiply $3$ by $- 1$ .

$\frac{9 x^{2} - 24 x + 64}{(3 x + 8) (9 x^{2} - 3 x \cdot 8 + 8^{2})}$

Step 1.4.4

Multiply $8$ by $- 3$ .

$\frac{9 x^{2} - 24 x + 64}{(3 x + 8) (9 x^{2} - 24 x + 8^{2})}$

Step 1.4.5

Raise $8$ to the power of $2$ .

$\frac{9 x^{2} - 24 x + 64}{(3 x + 8) (9 x^{2} - 24 x + 64)}$

Step 2

Cancel the common factor of $9 x^{2} - 24 x + 64$ .

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Step 2.1

Cancel the common factor.

$\frac{9 x^{2} - 24 x + 64}{(3 x + 8) (9 x^{2} - 24 x + 64)}$

Step 2.2

Rewrite the expression.

$\frac{1}{3 x + 8}$