Precalculus Examples

$\frac{\frac{7}{64} - \frac{7}{8 x}}{\frac{x}{64} - \frac{1}{x}}$

Step 1

Multiply the numerator and denominator of the fraction by $64 x$ .

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Step 1.1

Multiply $\frac{\frac{7}{64} - \frac{7}{8 x}}{\frac{x}{64} - \frac{1}{x}}$ by $\frac{64 x}{64 x}$ .

$\frac{64 x}{64 x} \cdot \frac{\frac{7}{64} - \frac{7}{8 x}}{\frac{x}{64} - \frac{1}{x}}$

Step 1.2

Combine.

$\frac{64 x (\frac{7}{64} - \frac{7}{8 x})}{64 x (\frac{x}{64} - \frac{1}{x})}$

Step 2

Apply the distributive property.

$\frac{64 x \frac{7}{64} + 64 x (- \frac{7}{8 x})}{64 x \frac{x}{64} + 64 x (- \frac{1}{x})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $64$ .

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Step 3.1.1

Factor $64$ out of $64 x$ .

$\frac{64 (x) \frac{7}{64} + 64 x (- \frac{7}{8 x})}{64 x \frac{x}{64} + 64 x (- \frac{1}{x})}$

Step 3.1.2

Cancel the common factor.

$\frac{64 x \frac{7}{64} + 64 x (- \frac{7}{8 x})}{64 x \frac{x}{64} + 64 x (- \frac{1}{x})}$

Step 3.1.3

Rewrite the expression.

$\frac{x \cdot 7 + 64 x (- \frac{7}{8 x})}{64 x \frac{x}{64} + 64 x (- \frac{1}{x})}$

Step 3.2

Cancel the common factor of $8 x$ .

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Step 3.2.1

Move the leading negative in $- \frac{7}{8 x}$ into the numerator.

$\frac{x \cdot 7 + 64 x \frac{- 7}{8 x}}{64 x \frac{x}{64} + 64 x (- \frac{1}{x})}$

Step 3.2.2

Factor $8 x$ out of $64 x$ .

$\frac{x \cdot 7 + 8 x (8) \frac{- 7}{8 x}}{64 x \frac{x}{64} + 64 x (- \frac{1}{x})}$

Step 3.2.3

Cancel the common factor.

$\frac{x \cdot 7 + 8 x \cdot 8 \frac{- 7}{8 x}}{64 x \frac{x}{64} + 64 x (- \frac{1}{x})}$

Step 3.2.4

Rewrite the expression.

$\frac{x \cdot 7 + 8 \cdot - 7}{64 x \frac{x}{64} + 64 x (- \frac{1}{x})}$

Step 3.3

Multiply $8$ by $- 7$ .

$\frac{x \cdot 7 - 56}{64 x \frac{x}{64} + 64 x (- \frac{1}{x})}$

Step 3.4

Cancel the common factor of $64$ .

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Step 3.4.1

Factor $64$ out of $64 x$ .

$\frac{x \cdot 7 - 56}{64 (x) \frac{x}{64} + 64 x (- \frac{1}{x})}$

Step 3.4.2

Cancel the common factor.

$\frac{x \cdot 7 - 56}{64 x \frac{x}{64} + 64 x (- \frac{1}{x})}$

Step 3.4.3

Rewrite the expression.

$\frac{x \cdot 7 - 56}{x \cdot x + 64 x (- \frac{1}{x})}$

Step 3.5

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

$\frac{x \cdot 7 - 56}{x^{1} x + 64 x (- \frac{1}{x})}$

Step 3.6

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

$\frac{x \cdot 7 - 56}{x^{1} x^{1} + 64 x (- \frac{1}{x})}$

Step 3.7

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

$\frac{x \cdot 7 - 56}{x^{1 + 1} + 64 x (- \frac{1}{x})}$

Step 3.8

Add $1$ and $1$ .

$\frac{x \cdot 7 - 56}{x^{2} + 64 x (- \frac{1}{x})}$

Step 3.9

Cancel the common factor of $x$ .

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Step 3.9.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$\frac{x \cdot 7 - 56}{x^{2} + 64 x \frac{- 1}{x}}$

Step 3.9.2

Factor $x$ out of $64 x$ .

$\frac{x \cdot 7 - 56}{x^{2} + x \cdot 64 \frac{- 1}{x}}$

Step 3.9.3

Cancel the common factor.

$\frac{x \cdot 7 - 56}{x^{2} + x \cdot 64 \frac{- 1}{x}}$

Step 3.9.4

Rewrite the expression.

$\frac{x \cdot 7 - 56}{x^{2} + 64 \cdot - 1}$

Step 3.10

Multiply $64$ by $- 1$ .

$\frac{x \cdot 7 - 56}{x^{2} - 64}$

Step 4

Factor $7$ out of $x \cdot 7 - 56$ .

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Step 4.1

Factor $7$ out of $x \cdot 7$ .

$\frac{7 \cdot x - 56}{x^{2} - 64}$

Step 4.2

Factor $7$ out of $- 56$ .

$\frac{7 \cdot x + 7 \cdot - 8}{x^{2} - 64}$

Step 4.3

Factor $7$ out of $7 \cdot x + 7 \cdot - 8$ .

$\frac{7 (x - 8)}{x^{2} - 64}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $64$ as $8^{2}$ .

$\frac{7 (x - 8)}{x^{2} - 8^{2}}$

Step 5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 8$ .

$\frac{7 (x - 8)}{(x + 8) (x - 8)}$

Step 6

Cancel the common factor of $x - 8$ .

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Step 6.1

Cancel the common factor.

$\frac{7 (x - 8)}{(x + 8) (x - 8)}$

Step 6.2

Rewrite the expression.

$\frac{7}{x + 8}$