Algebra Examples

$\frac{4 x^{3} {(x^{2} - 4)}^{\frac{5}{3}}}{{(2 x)}^{3} (x - 2) \sqrt[3]{{(x^{2} - 4)}^{2}}}$

Step 1

Simplify the denominator.

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Step 1.1

Apply the product rule to $2 x$ .

$\frac{4 x^{3} {(x^{2} - 4)}^{\frac{5}{3}}}{2^{3} x^{3} (x - 2) \sqrt[3]{{(x^{2} - 4)}^{2}}}$

Step 1.2

Raise $2$ to the power of $3$ .

$\frac{4 x^{3} {(x^{2} - 4)}^{\frac{5}{3}}}{8 x^{3} (x - 2) \sqrt[3]{{(x^{2} - 4)}^{2}}}$

Step 1.3

Rewrite $4$ as $2^{2}$ .

$\frac{4 x^{3} {(x^{2} - 4)}^{\frac{5}{3}}}{8 x^{3} (x - 2) \sqrt[3]{{(x^{2} - 2^{2})}^{2}}}$

Step 1.4

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 $b = 2$ .

$\frac{4 x^{3} {(x^{2} - 4)}^{\frac{5}{3}}}{8 x^{3} (x - 2) \sqrt[3]{{((x + 2) (x - 2))}^{2}}}$

Step 1.5

Apply the product rule to $(x + 2) (x - 2)$ .

$\frac{4 x^{3} {(x^{2} - 4)}^{\frac{5}{3}}}{8 x^{3} (x - 2) \sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}$

Step 2

Factor $4$ out of $4 x^{3} {(x^{2} - 4)}^{\frac{5}{3}}$ .

$\frac{4 (x^{3} {(x^{2} - 4)}^{\frac{5}{3}})}{8 x^{3} (x - 2) \sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}$

Step 3

Cancel the common factors.

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Step 3.1

Factor $4$ out of $8 x^{3} (x - 2) \sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}$ .

$\frac{4 (x^{3} {(x^{2} - 4)}^{\frac{5}{3}})}{4 (2 x^{3} (x - 2) \sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}})}$

Step 3.2

Cancel the common factor.

$\frac{4 (x^{3} {(x^{2} - 4)}^{\frac{5}{3}})}{4 (2 x^{3} (x - 2) \sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}})}$

Step 3.3

Rewrite the expression.

$\frac{x^{3} {(x^{2} - 4)}^{\frac{5}{3}}}{2 x^{3} (x - 2) \sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}$

Step 4

Cancel the common factor.

$\frac{x^{3} {(x^{2} - 4)}^{\frac{5}{3}}}{2 x^{3} (x - 2) \sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}$

Step 5

Rewrite the expression.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}}}{(2 (x - 2)) \sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}$

Step 6

Multiply $\frac{{(x^{2} - 4)}^{\frac{5}{3}}}{(2 (x - 2)) \sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}$ by $\frac{{\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{{\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}}}{(2 (x - 2)) \sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}} \cdot \frac{{\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{{\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{{(x^{2} - 4)}^{\frac{5}{3}}}{(2 (x - 2)) \sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}$ by $\frac{{\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{{\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{(2 (x - 2)) \sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}$

Step 7.2

Move $\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{2 (x - 2) (\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2})}$

Step 7.3

Raise $\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}$ to the power of $1$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{2 (x - 2) ({\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{1} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2})}$

Step 7.4

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

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{2 (x - 2) {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{1 + 2}}$

Step 7.5

Add $1$ and $2$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{2 (x - 2) {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{3}}$

Step 7.6

Rewrite ${\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{3}$ as ${(x + 2)}^{2} {(x - 2)}^{2}$ .

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Step 7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}$ as ${({(x + 2)}^{2} {(x - 2)}^{2})}^{\frac{1}{3}}$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{2 (x - 2) {({({(x + 2)}^{2} {(x - 2)}^{2})}^{\frac{1}{3}})}^{3}}$

Step 7.6.2

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

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{2 (x - 2) {({(x + 2)}^{2} {(x - 2)}^{2})}^{\frac{1}{3} \cdot 3}}$

Step 7.6.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{2 (x - 2) {({(x + 2)}^{2} {(x - 2)}^{2})}^{\frac{3}{3}}}$

Step 7.6.4

Cancel the common factor of $3$ .

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Step 7.6.4.1

Cancel the common factor.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{2 (x - 2) {({(x + 2)}^{2} {(x - 2)}^{2})}^{\frac{3}{3}}}$

Step 7.6.4.2

Rewrite the expression.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{2 (x - 2) {({(x + 2)}^{2} {(x - 2)}^{2})}^{1}}$

Step 7.6.5

Simplify.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{2 (x - 2) ({(x + 2)}^{2} {(x - 2)}^{2})}$

Step 8

Multiply $x - 2$ by ${(x - 2)}^{2}$ by adding the exponents.

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Step 8.1

Move ${(x - 2)}^{2}$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{2 ({(x - 2)}^{2} (x - 2)) {(x + 2)}^{2}}$

Step 8.2

Multiply ${(x - 2)}^{2}$ by $x - 2$ .

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Step 8.2.1

Raise $x - 2$ to the power of $1$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{2 ({(x - 2)}^{2} {(x - 2)}^{1}) {(x + 2)}^{2}}$

Step 8.2.2

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

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{2 {(x - 2)}^{2 + 1} {(x + 2)}^{2}}$

Step 8.3

Add $2$ and $1$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} {\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9

Simplify the numerator.

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Step 9.1

Rewrite ${\sqrt[3]{{(x + 2)}^{2} {(x - 2)}^{2}}}^{2}$ as $\sqrt[3]{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{{({(x + 2)}^{2} {(x - 2)}^{2})}^{2}}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.2

Apply the product rule to ${(x + 2)}^{2} {(x - 2)}^{2}$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{{({(x + 2)}^{2})}^{2} {({(x - 2)}^{2})}^{2}}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.3

Multiply the exponents in ${({(x + 2)}^{2})}^{2}$ .

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Step 9.3.1

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

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{{(x + 2)}^{2 \cdot 2} {({(x - 2)}^{2})}^{2}}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.3.2

Multiply $2$ by $2$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{{(x + 2)}^{4} {({(x - 2)}^{2})}^{2}}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.4

Use the Binomial Theorem.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x^{4} + 4 x^{3} \cdot 2 + 6 x^{2} \cdot 2^{2} + 4 x \cdot 2^{3} + 2^{4}) {({(x - 2)}^{2})}^{2}}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.5

Simplify each term.

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Step 9.5.1

Multiply $2$ by $4$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x^{4} + 8 x^{3} + 6 x^{2} \cdot 2^{2} + 4 x \cdot 2^{3} + 2^{4}) {({(x - 2)}^{2})}^{2}}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.5.2

Raise $2$ to the power of $2$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x^{4} + 8 x^{3} + 6 x^{2} \cdot 4 + 4 x \cdot 2^{3} + 2^{4}) {({(x - 2)}^{2})}^{2}}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.5.3

Multiply $4$ by $6$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x^{4} + 8 x^{3} + 24 x^{2} + 4 x \cdot 2^{3} + 2^{4}) {({(x - 2)}^{2})}^{2}}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.5.4

Raise $2$ to the power of $3$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x^{4} + 8 x^{3} + 24 x^{2} + 4 x \cdot 8 + 2^{4}) {({(x - 2)}^{2})}^{2}}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.5.5

Multiply $8$ by $4$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 2^{4}) {({(x - 2)}^{2})}^{2}}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.5.6

Raise $2$ to the power of $4$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) {({(x - 2)}^{2})}^{2}}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.6

Multiply the exponents in ${({(x - 2)}^{2})}^{2}$ .

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Step 9.6.1

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

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) {(x - 2)}^{2 \cdot 2}}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.6.2

Multiply $2$ by $2$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) {(x - 2)}^{4}}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.7

Use the Binomial Theorem.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) (x^{4} + 4 x^{3} \cdot - 2 + 6 x^{2} {(- 2)}^{2} + 4 x {(- 2)}^{3} + {(- 2)}^{4})}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.8

Simplify each term.

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Step 9.8.1

Multiply $- 2$ by $4$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) (x^{4} - 8 x^{3} + 6 x^{2} {(- 2)}^{2} + 4 x {(- 2)}^{3} + {(- 2)}^{4})}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.8.2

Raise $- 2$ to the power of $2$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) (x^{4} - 8 x^{3} + 6 x^{2} \cdot 4 + 4 x {(- 2)}^{3} + {(- 2)}^{4})}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.8.3

Multiply $4$ by $6$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) (x^{4} - 8 x^{3} + 24 x^{2} + 4 x {(- 2)}^{3} + {(- 2)}^{4})}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.8.4

Raise $- 2$ to the power of $3$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) (x^{4} - 8 x^{3} + 24 x^{2} + 4 x \cdot - 8 + {(- 2)}^{4})}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.8.5

Multiply $- 8$ by $4$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) (x^{4} - 8 x^{3} + 24 x^{2} - 32 x + {(- 2)}^{4})}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.8.6

Raise $- 2$ to the power of $4$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) (x^{4} - 8 x^{3} + 24 x^{2} - 32 x + 16)}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.9

Make each term match the terms from the binomial theorem formula.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x^{4} + 4 \cdot 2 x^{3} + 6 \cdot 2^{2} x^{2} + 4 \cdot 2^{3} x + 2^{4}) (x^{4} - 8 x^{3} + 24 x^{2} - 32 x + 16)}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.10

Factor $x^{4} + 4 \cdot 2 x^{3} + 6 \cdot 2^{2} x^{2} + 4 \cdot 2^{3} x + 2^{4}$ using the binomial theorem.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{{(x + 2)}^{4} (x^{4} - 8 x^{3} + 24 x^{2} - 32 x + 16)}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.11

Make each term match the terms from the binomial theorem formula.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{{(x + 2)}^{4} (x^{4} - 4 \cdot 2 x^{3} + 6 \cdot 2^{2} x^{2} - 4 \cdot 2^{3} x + 2^{4})}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.12

Factor $x^{4} - 4 \cdot 2 x^{3} + 6 \cdot 2^{2} x^{2} - 4 \cdot 2^{3} x + 2^{4}$ using the binomial theorem.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{{(x + 2)}^{4} {(2 - x)}^{4}}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.13

Rewrite ${(x + 2)}^{4} {(2 - x)}^{4}$ as ${((x + 2) (2 - x))}^{3} ((x + 2) (2 - x))$ .

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Step 9.13.1

Factor out ${(x + 2)}^{3}$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{{(x + 2)}^{3} (x + 2) {(2 - x)}^{4}}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.13.2

Factor out ${(2 - x)}^{3}$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{{(x + 2)}^{3} (x + 2) ({(2 - x)}^{3} (2 - x))}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.13.3

Move $x + 2$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{{(x + 2)}^{3} {(2 - x)}^{3} (x + 2) (2 - x)}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.13.4

Rewrite ${(x + 2)}^{3} {(2 - x)}^{3}$ as ${((x + 2) (2 - x))}^{3}$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{{((x + 2) (2 - x))}^{3} (x + 2) (2 - x)}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.13.5

Add parentheses.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{{((x + 2) (2 - x))}^{3} ((x + 2) (2 - x))}}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.14

Pull terms out from under the radical.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} ((x + 2) (2 - x) \sqrt[3]{(x + 2) (2 - x)})}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.15

Expand $(x + 2) (2 - x)$ using the FOIL Method.

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Step 9.15.1

Apply the distributive property.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} ((x (2 - x) + 2 (2 - x)) \sqrt[3]{(x + 2) (2 - x)})}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.15.2

Apply the distributive property.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} ((x \cdot 2 + x (- x) + 2 (2 - x)) \sqrt[3]{(x + 2) (2 - x)})}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.15.3

Apply the distributive property.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} ((x \cdot 2 + x (- x) + 2 \cdot 2 + 2 (- x)) \sqrt[3]{(x + 2) (2 - x)})}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.16

Simplify and combine like terms.

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Step 9.16.1

Simplify each term.

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Step 9.16.1.1

Move $2$ to the left of $x$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} ((2 \cdot x + x (- x) + 2 \cdot 2 + 2 (- x)) \sqrt[3]{(x + 2) (2 - x)})}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.16.1.2

Rewrite using the commutative property of multiplication.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} ((2 x - x \cdot x + 2 \cdot 2 + 2 (- x)) \sqrt[3]{(x + 2) (2 - x)})}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.16.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 9.16.1.3.1

Move $x$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} ((2 x - (x \cdot x) + 2 \cdot 2 + 2 (- x)) \sqrt[3]{(x + 2) (2 - x)})}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.16.1.3.2

Multiply $x$ by $x$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} ((2 x - x^{2} + 2 \cdot 2 + 2 (- x)) \sqrt[3]{(x + 2) (2 - x)})}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.16.1.4

Multiply $2$ by $2$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} ((2 x - x^{2} + 4 + 2 (- x)) \sqrt[3]{(x + 2) (2 - x)})}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.16.1.5

Multiply $- 1$ by $2$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} ((2 x - x^{2} + 4 - 2 x) \sqrt[3]{(x + 2) (2 - x)})}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.16.2

Subtract $2 x$ from $2 x$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} ((- x^{2} + 4 + 0) \sqrt[3]{(x + 2) (2 - x)})}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.16.3

Add $- x^{2}$ and $0$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} ((- x^{2} + 4) \sqrt[3]{(x + 2) (2 - x)})}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.17

Apply the distributive property.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} (- x^{2} \sqrt[3]{(x + 2) (2 - x)} + 4 \sqrt[3]{(x + 2) (2 - x)})}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.18

Factor $\sqrt[3]{(x + 2) (2 - x)}$ out of $- x^{2} \sqrt[3]{(x + 2) (2 - x)} + 4 \sqrt[3]{(x + 2) (2 - x)}$ .

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Step 9.18.1

Factor $\sqrt[3]{(x + 2) (2 - x)}$ out of $- x^{2} \sqrt[3]{(x + 2) (2 - x)}$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} (\sqrt[3]{(x + 2) (2 - x)} (- x^{2}) + 4 \sqrt[3]{(x + 2) (2 - x)})}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.18.2

Factor $\sqrt[3]{(x + 2) (2 - x)}$ out of $4 \sqrt[3]{(x + 2) (2 - x)}$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} (\sqrt[3]{(x + 2) (2 - x)} (- x^{2}) + \sqrt[3]{(x + 2) (2 - x)} \cdot 4)}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.18.3

Factor $\sqrt[3]{(x + 2) (2 - x)}$ out of $\sqrt[3]{(x + 2) (2 - x)} (- x^{2}) + \sqrt[3]{(x + 2) (2 - x)} \cdot 4$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} (\sqrt[3]{(x + 2) (2 - x)} (- x^{2} + 4))}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.19

Rewrite $4$ as $2^{2}$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} (\sqrt[3]{(x + 2) (2 - x)} (- x^{2} + 2^{2}))}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.20

Reorder $- x^{2}$ and $2^{2}$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} (\sqrt[3]{(x + 2) (2 - x)} (2^{2} - x^{2}))}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 9.21

Factor.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)} (2 + x) (2 - x)}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 10

Simplify with factoring out.

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Step 10.1

Reorder terms.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)} (x + 2) (2 - x)}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 10.2

Factor $x + 2$ out of ${(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)} (x + 2) (2 - x)$ .

$\frac{(x + 2) ({(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)} (2 - x))}{2 {(x - 2)}^{3} {(x + 2)}^{2}}$

Step 11

Cancel the common factors.

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Step 11.1

Factor $x + 2$ out of $2 {(x - 2)}^{3} {(x + 2)}^{2}$ .

$\frac{(x + 2) ({(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)} (2 - x))}{(x + 2) (2 {(x - 2)}^{3} (x + 2))}$

Step 11.2

Cancel the common factor.

$\frac{(x + 2) ({(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)} (2 - x))}{(x + 2) (2 {(x - 2)}^{3} (x + 2))}$

Step 11.3

Rewrite the expression.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)} (2 - x)}{2 {(x - 2)}^{3} (x + 2)}$

Step 12

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)} (- 1 (- 2) - x)}{2 {(x - 2)}^{3} (x + 2)}$

Step 13

Factor $- 1$ out of $- x$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)} (- 1 (- 2) - (x))}{2 {(x - 2)}^{3} (x + 2)}$

Step 14

Factor $- 1$ out of $- 1 (- 2) - (x)$ .

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)} (- 1 (- 2 + x))}{2 {(x - 2)}^{3} (x + 2)}$

Step 15

Reorder terms.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)} (- 1 (x - 2))}{2 {(x - 2)}^{3} (x + 2)}$

Step 16

Factor $x - 2$ out of ${(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)} (- 1 (x - 2))$ .

$\frac{(x - 2) ({(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)} \cdot (- 1))}{2 {(x - 2)}^{3} (x + 2)}$

Step 17

Cancel the common factors.

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Step 17.1

Factor $x - 2$ out of $2 {(x - 2)}^{3} (x + 2)$ .

$\frac{(x - 2) ({(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)} \cdot (- 1))}{(x - 2) (2 {(x - 2)}^{2} (x + 2))}$

Step 17.2

Cancel the common factor.

$\frac{(x - 2) ({(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)} \cdot (- 1))}{(x - 2) (2 {(x - 2)}^{2} (x + 2))}$

Step 17.3

Rewrite the expression.

$\frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)} \cdot (- 1)}{2 {(x - 2)}^{2} (x + 2)}$

Step 18

Move $- 1$ to the left of ${(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)}$ .

$\frac{- 1 \cdot ({(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)})}{2 {(x - 2)}^{2} (x + 2)}$

Step 19

Move the negative in front of the fraction.

$- \frac{{(x^{2} - 4)}^{\frac{5}{3}} \sqrt[3]{(x + 2) (2 - x)}}{2 {(x - 2)}^{2} (x + 2)}$