Precalculus Examples

$\frac{\frac{x + 1}{x^{3}} + 1}{{(\frac{x + 1}{x^{3}})}^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

Write $1$ as a fraction with a common denominator.

$\frac{\frac{x + 1}{x^{3}} + \frac{x^{3}}{x^{3}}}{{(\frac{x + 1}{x^{3}})}^{3}}$

Step 1.2

Combine the numerators over the common denominator.

$\frac{\frac{x + 1 + x^{3}}{x^{3}}}{{(\frac{x + 1}{x^{3}})}^{3}}$

Step 1.3

Reorder terms.

$\frac{\frac{x^{3} + x + 1}{x^{3}}}{{(\frac{x + 1}{x^{3}})}^{3}}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $\frac{x + 1}{x^{3}}$ .

$\frac{\frac{x^{3} + x + 1}{x^{3}}}{\frac{{(x + 1)}^{3}}{{(x^{3})}^{3}}}$

Step 2.2

Multiply the exponents in ${(x^{3})}^{3}$ .

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Step 2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{x^{3} + x + 1}{x^{3}}}{\frac{{(x + 1)}^{3}}{x^{3 \cdot 3}}}$

Step 2.2.2

Multiply $3$ by $3$ .

$\frac{\frac{x^{3} + x + 1}{x^{3}}}{\frac{{(x + 1)}^{3}}{x^{9}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{3} + x + 1}{x^{3}} \cdot \frac{x^{9}}{{(x + 1)}^{3}}$

Step 4

Cancel the common factor of $x^{3}$ .

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Step 4.1

Factor $x^{3}$ out of $x^{9}$ .

$\frac{x^{3} + x + 1}{x^{3}} \cdot \frac{x^{3} x^{6}}{{(x + 1)}^{3}}$

Step 4.2

Cancel the common factor.

$\frac{x^{3} + x + 1}{x^{3}} \cdot \frac{x^{3} x^{6}}{{(x + 1)}^{3}}$

Step 4.3

Rewrite the expression.

$(x^{3} + x + 1) \frac{x^{6}}{{(x + 1)}^{3}}$

Step 5

Multiply $x^{3} + x + 1$ by $\frac{x^{6}}{{(x + 1)}^{3}}$ .

$\frac{(x^{3} + x + 1) x^{6}}{{(x + 1)}^{3}}$

Step 6

Reorder factors in $\frac{(x^{3} + x + 1) x^{6}}{{(x + 1)}^{3}}$ .

$\frac{x^{6} (x^{3} + x + 1)}{{(x + 1)}^{3}}$