Precalculus Examples

$\frac{- 5 m {(m + 4)}^{- \frac{2}{3}} + 6 {(m + 4)}^{\frac{1}{3}}}{m + 4}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{- 5 m \frac{1}{{(m + 4)}^{\frac{2}{3}}} + 6 {(m + 4)}^{\frac{1}{3}}}{m + 4}$

Step 1.2

Multiply $- 5 m \frac{1}{{(m + 4)}^{\frac{2}{3}}}$ .

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Step 1.2.1

Combine $- 5$ and $\frac{1}{{(m + 4)}^{\frac{2}{3}}}$ .

$\frac{m \frac{- 5}{{(m + 4)}^{\frac{2}{3}}} + 6 {(m + 4)}^{\frac{1}{3}}}{m + 4}$

Step 1.2.2

Combine $m$ and $\frac{- 5}{{(m + 4)}^{\frac{2}{3}}}$ .

$\frac{\frac{m \cdot - 5}{{(m + 4)}^{\frac{2}{3}}} + 6 {(m + 4)}^{\frac{1}{3}}}{m + 4}$

Step 1.3

Move $- 5$ to the left of $m$ .

$\frac{\frac{- 5 \cdot m}{{(m + 4)}^{\frac{2}{3}}} + 6 {(m + 4)}^{\frac{1}{3}}}{m + 4}$

Step 1.4

Move the negative in front of the fraction.

$\frac{- \frac{5 \cdot m}{{(m + 4)}^{\frac{2}{3}}} + 6 {(m + 4)}^{\frac{1}{3}}}{m + 4}$

Step 1.5

To write $6 {(m + 4)}^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{{(m + 4)}^{\frac{2}{3}}}{{(m + 4)}^{\frac{2}{3}}}$ .

$\frac{- \frac{5 m}{{(m + 4)}^{\frac{2}{3}}} + \frac{6 {(m + 4)}^{\frac{1}{3}} {(m + 4)}^{\frac{2}{3}}}{{(m + 4)}^{\frac{2}{3}}}}{m + 4}$

Step 1.6

Combine the numerators over the common denominator.

$\frac{\frac{- 5 m + 6 {(m + 4)}^{\frac{1}{3}} {(m + 4)}^{\frac{2}{3}}}{{(m + 4)}^{\frac{2}{3}}}}{m + 4}$

Step 1.7

Reorder terms.

$\frac{\frac{6 {(m + 4)}^{\frac{2}{3}} {(m + 4)}^{\frac{1}{3}} - 5 m}{{(m + 4)}^{\frac{2}{3}}}}{m + 4}$

Step 1.8

Rewrite $\frac{6 {(m + 4)}^{\frac{2}{3}} {(m + 4)}^{\frac{1}{3}} - 5 m}{{(m + 4)}^{\frac{2}{3}}}$ in a factored form.

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Step 1.8.1

Multiply ${(m + 4)}^{\frac{2}{3}}$ by ${(m + 4)}^{\frac{1}{3}}$ by adding the exponents.

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Step 1.8.1.1

Move ${(m + 4)}^{\frac{1}{3}}$ .

$\frac{\frac{6 ({(m + 4)}^{\frac{1}{3}} {(m + 4)}^{\frac{2}{3}}) - 5 m}{{(m + 4)}^{\frac{2}{3}}}}{m + 4}$

Step 1.8.1.2

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

$\frac{\frac{6 {(m + 4)}^{\frac{1}{3} + \frac{2}{3}} - 5 m}{{(m + 4)}^{\frac{2}{3}}}}{m + 4}$

Step 1.8.1.3

Combine the numerators over the common denominator.

$\frac{\frac{6 {(m + 4)}^{\frac{1 + 2}{3}} - 5 m}{{(m + 4)}^{\frac{2}{3}}}}{m + 4}$

Step 1.8.1.4

Add $1$ and $2$ .

$\frac{\frac{6 {(m + 4)}^{\frac{3}{3}} - 5 m}{{(m + 4)}^{\frac{2}{3}}}}{m + 4}$

Step 1.8.1.5

Divide $3$ by $3$ .

$\frac{\frac{6 {(m + 4)}^{1} - 5 m}{{(m + 4)}^{\frac{2}{3}}}}{m + 4}$

Step 1.8.2

Simplify $6 {(m + 4)}^{1}$ .

$\frac{\frac{6 (m + 4) - 5 m}{{(m + 4)}^{\frac{2}{3}}}}{m + 4}$

Step 1.8.3

Apply the distributive property.

$\frac{\frac{6 m + 6 \cdot 4 - 5 m}{{(m + 4)}^{\frac{2}{3}}}}{m + 4}$

Step 1.8.4

Multiply $6$ by $4$ .

$\frac{\frac{6 m + 24 - 5 m}{{(m + 4)}^{\frac{2}{3}}}}{m + 4}$

Step 1.8.5

Subtract $5 m$ from $6 m$ .

$\frac{\frac{m + 24}{{(m + 4)}^{\frac{2}{3}}}}{m + 4}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{m + 24}{{(m + 4)}^{\frac{2}{3}}} \cdot \frac{1}{m + 4}$

Step 3

Multiply $\frac{m + 24}{{(m + 4)}^{\frac{2}{3}}} \cdot \frac{1}{m + 4}$ .

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Step 3.1

Multiply $\frac{m + 24}{{(m + 4)}^{\frac{2}{3}}}$ by $\frac{1}{m + 4}$ .

$\frac{m + 24}{{(m + 4)}^{\frac{2}{3}} (m + 4)}$

Step 3.2

Multiply ${(m + 4)}^{\frac{2}{3}}$ by $m + 4$ by adding the exponents.

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Step 3.2.1

Multiply ${(m + 4)}^{\frac{2}{3}}$ by $m + 4$ .

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Step 3.2.1.1

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

$\frac{m + 24}{{(m + 4)}^{\frac{2}{3}} {(m + 4)}^{1}}$

Step 3.2.1.2

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

$\frac{m + 24}{{(m + 4)}^{\frac{2}{3} + 1}}$

Step 3.2.2

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

$\frac{m + 24}{{(m + 4)}^{\frac{2}{3} + \frac{3}{3}}}$

Step 3.2.3

Combine the numerators over the common denominator.

$\frac{m + 24}{{(m + 4)}^{\frac{2 + 3}{3}}}$

Step 3.2.4

Add $2$ and $3$ .

$\frac{m + 24}{{(m + 4)}^{\frac{5}{3}}}$