Algebra Examples

$\frac{8 {(9 + x)}^{\frac{1}{3}} - x {(9 + x)}^{- \frac{2}{3}}}{{(9 + x)}^{\frac{2}{3}}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{8 {(9 + x)}^{\frac{1}{3}} - x \frac{1}{{(9 + x)}^{\frac{2}{3}}}}{{(9 + x)}^{\frac{2}{3}}}$

Step 1.2

Combine $\frac{1}{{(9 + x)}^{\frac{2}{3}}}$ and $x$ .

$\frac{8 {(9 + x)}^{\frac{1}{3}} - \frac{x}{{(9 + x)}^{\frac{2}{3}}}}{{(9 + x)}^{\frac{2}{3}}}$

Step 1.3

To write $8 {(9 + x)}^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{{(9 + x)}^{\frac{2}{3}}}{{(9 + x)}^{\frac{2}{3}}}$ .

$\frac{\frac{8 {(9 + x)}^{\frac{1}{3}} {(9 + x)}^{\frac{2}{3}}}{{(9 + x)}^{\frac{2}{3}}} - \frac{x}{{(9 + x)}^{\frac{2}{3}}}}{{(9 + x)}^{\frac{2}{3}}}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{\frac{8 {(9 + x)}^{\frac{1}{3}} {(9 + x)}^{\frac{2}{3}} - x}{{(9 + x)}^{\frac{2}{3}}}}{{(9 + x)}^{\frac{2}{3}}}$

Step 1.5

Reorder terms.

$\frac{\frac{8 {(9 + x)}^{\frac{2}{3}} {(9 + x)}^{\frac{1}{3}} - x}{{(x + 9)}^{\frac{2}{3}}}}{{(9 + x)}^{\frac{2}{3}}}$

Step 1.6

Rewrite $\frac{8 {(9 + x)}^{\frac{2}{3}} {(9 + x)}^{\frac{1}{3}} - x}{{(x + 9)}^{\frac{2}{3}}}$ in a factored form.

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Step 1.6.1

Multiply ${(9 + x)}^{\frac{2}{3}}$ by ${(9 + x)}^{\frac{1}{3}}$ by adding the exponents.

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Step 1.6.1.1

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

$\frac{\frac{8 ({(9 + x)}^{\frac{1}{3}} {(9 + x)}^{\frac{2}{3}}) - x}{{(x + 9)}^{\frac{2}{3}}}}{{(9 + x)}^{\frac{2}{3}}}$

Step 1.6.1.2

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

$\frac{\frac{8 {(9 + x)}^{\frac{1}{3} + \frac{2}{3}} - x}{{(x + 9)}^{\frac{2}{3}}}}{{(9 + x)}^{\frac{2}{3}}}$

Step 1.6.1.3

Combine the numerators over the common denominator.

$\frac{\frac{8 {(9 + x)}^{\frac{1 + 2}{3}} - x}{{(x + 9)}^{\frac{2}{3}}}}{{(9 + x)}^{\frac{2}{3}}}$

Step 1.6.1.4

Add $1$ and $2$ .

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

Step 1.6.1.5

Divide $3$ by $3$ .

$\frac{\frac{8 {(9 + x)}^{1} - x}{{(x + 9)}^{\frac{2}{3}}}}{{(9 + x)}^{\frac{2}{3}}}$

Step 1.6.2

Simplify $8 {(9 + x)}^{1}$ .

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

Step 1.6.3

Apply the distributive property.

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

Step 1.6.4

Multiply $8$ by $9$ .

$\frac{\frac{72 + 8 x - x}{{(x + 9)}^{\frac{2}{3}}}}{{(9 + x)}^{\frac{2}{3}}}$

Step 1.6.5

Subtract $x$ from $8 x$ .

$\frac{\frac{72 + 7 x}{{(x + 9)}^{\frac{2}{3}}}}{{(9 + x)}^{\frac{2}{3}}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{72 + 7 x}{{(x + 9)}^{\frac{2}{3}}} \cdot \frac{1}{{(9 + x)}^{\frac{2}{3}}}$

Step 3

Multiply $\frac{72 + 7 x}{{(x + 9)}^{\frac{2}{3}}} \cdot \frac{1}{{(9 + x)}^{\frac{2}{3}}}$ .

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Step 3.1

Multiply $\frac{72 + 7 x}{{(x + 9)}^{\frac{2}{3}}}$ by $\frac{1}{{(9 + x)}^{\frac{2}{3}}}$ .

$\frac{72 + 7 x}{{(x + 9)}^{\frac{2}{3}} {(9 + x)}^{\frac{2}{3}}}$

Step 3.2

Reorder terms.

$\frac{72 + 7 x}{{(x + 9)}^{\frac{2}{3}} {(x + 9)}^{\frac{2}{3}}}$

Step 3.3

Multiply ${(x + 9)}^{\frac{2}{3}}$ by ${(x + 9)}^{\frac{2}{3}}$ by adding the exponents.

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Step 3.3.1

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

$\frac{72 + 7 x}{{(x + 9)}^{\frac{2}{3} + \frac{2}{3}}}$

Step 3.3.2

Combine the numerators over the common denominator.

$\frac{72 + 7 x}{{(x + 9)}^{\frac{2 + 2}{3}}}$

Step 3.3.3

Add $2$ and $2$ .

$\frac{72 + 7 x}{{(x + 9)}^{\frac{4}{3}}}$