Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

Factor $x$ out of $x {(x + 4)}^{- \frac{3}{4}}$ .

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

Step 1.1.2

Factor $x$ out of $x^{2} {(x + 4)}^{\frac{1}{4}}$ .

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

Step 1.1.3

Factor $x$ out of $x ({(x + 4)}^{- \frac{3}{4}}) + x (x {(x + 4)}^{\frac{1}{4}})$ .

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

Step 1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{x (\frac{1}{{(x + 4)}^{\frac{3}{4}}} + x {(x + 4)}^{\frac{1}{4}})}{x (x + 4)}$

Step 1.3

To write $x {(x + 4)}^{\frac{1}{4}}$ as a fraction with a common denominator, multiply by $\frac{{(x + 4)}^{\frac{3}{4}}}{{(x + 4)}^{\frac{3}{4}}}$ .

$\frac{x (\frac{1}{{(x + 4)}^{\frac{3}{4}}} + \frac{x {(x + 4)}^{\frac{1}{4}} {(x + 4)}^{\frac{3}{4}}}{{(x + 4)}^{\frac{3}{4}}})}{x (x + 4)}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{x \frac{1 + x {(x + 4)}^{\frac{1}{4}} {(x + 4)}^{\frac{3}{4}}}{{(x + 4)}^{\frac{3}{4}}}}{x (x + 4)}$

Step 1.5

Reorder terms.

$\frac{x \frac{x {(x + 4)}^{\frac{3}{4}} {(x + 4)}^{\frac{1}{4}} + 1}{{(x + 4)}^{\frac{3}{4}}}}{x (x + 4)}$

Step 1.6

Rewrite $\frac{x {(x + 4)}^{\frac{3}{4}} {(x + 4)}^{\frac{1}{4}} + 1}{{(x + 4)}^{\frac{3}{4}}}$ in a factored form.

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Step 1.6.1

Multiply ${(x + 4)}^{\frac{3}{4}}$ by ${(x + 4)}^{\frac{1}{4}}$ by adding the exponents.

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Step 1.6.1.1

Move ${(x + 4)}^{\frac{1}{4}}$ .

$\frac{x \frac{x ({(x + 4)}^{\frac{1}{4}} {(x + 4)}^{\frac{3}{4}}) + 1}{{(x + 4)}^{\frac{3}{4}}}}{x (x + 4)}$

Step 1.6.1.2

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

$\frac{x \frac{x {(x + 4)}^{\frac{1}{4} + \frac{3}{4}} + 1}{{(x + 4)}^{\frac{3}{4}}}}{x (x + 4)}$

Step 1.6.1.3

Combine the numerators over the common denominator.

$\frac{x \frac{x {(x + 4)}^{\frac{1 + 3}{4}} + 1}{{(x + 4)}^{\frac{3}{4}}}}{x (x + 4)}$

Step 1.6.1.4

Add $1$ and $3$ .

$\frac{x \frac{x {(x + 4)}^{\frac{4}{4}} + 1}{{(x + 4)}^{\frac{3}{4}}}}{x (x + 4)}$

Step 1.6.1.5

Divide $4$ by $4$ .

$\frac{x \frac{x {(x + 4)}^{1} + 1}{{(x + 4)}^{\frac{3}{4}}}}{x (x + 4)}$

Step 1.6.2

Simplify $x {(x + 4)}^{1}$ .

$\frac{x \frac{x (x + 4) + 1}{{(x + 4)}^{\frac{3}{4}}}}{x (x + 4)}$

Step 1.6.3

Apply the distributive property.

$\frac{x \frac{x \cdot x + x \cdot 4 + 1}{{(x + 4)}^{\frac{3}{4}}}}{x (x + 4)}$

Step 1.6.4

Multiply $x$ by $x$ .

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

Step 1.6.5

Move $4$ to the left of $x$ .

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

Step 2

Combine $x$ and $\frac{x^{2} + 4 x + 1}{{(x + 4)}^{\frac{3}{4}}}$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Combine.

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

Step 5

Multiply ${(x + 4)}^{\frac{3}{4}}$ by $x + 4$ by adding the exponents.

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Step 5.1

Move $x + 4$ .

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

Step 5.2

Multiply $x + 4$ by ${(x + 4)}^{\frac{3}{4}}$ .

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Step 5.2.1

Raise $x + 4$ to the power of $1$ .

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

Step 5.2.2

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

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

Step 5.3

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

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

Step 5.4

Combine the numerators over the common denominator.

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

Step 5.5

Add $4$ and $3$ .

$\frac{x (x^{2} + 4 x + 1) \cdot 1}{{(x + 4)}^{\frac{7}{4}} x}$

Step 6

Multiply $x (x^{2} + 4 x + 1)$ by $1$ .

$\frac{x (x^{2} + 4 x + 1)}{{(x + 4)}^{\frac{7}{4}} x}$

Step 7

Cancel the common factor.

$\frac{x (x^{2} + 4 x + 1)}{{(x + 4)}^{\frac{7}{4}} x}$

Step 8

Rewrite the expression.

$\frac{x^{2} + 4 x + 1}{{(x + 4)}^{\frac{7}{4}}}$