Algebra Examples

${(- 2 a^{\frac{2}{3}} b)}^{3} {(a b^{\frac{1}{3}})}^{3}$

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1

Apply the product rule to $- 2 a^{\frac{2}{3}} b$ .

${(- 2 a^{\frac{2}{3}})}^{3} b^{3} {(a b^{\frac{1}{3}})}^{3}$

Step 1.2

Apply the product rule to $- 2 a^{\frac{2}{3}}$ .

${(- 2)}^{3} {(a^{\frac{2}{3}})}^{3} b^{3} {(a b^{\frac{1}{3}})}^{3}$

Step 2

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

$- 8 {(a^{\frac{2}{3}})}^{3} b^{3} {(a b^{\frac{1}{3}})}^{3}$

Step 3

Multiply the exponents in ${(a^{\frac{2}{3}})}^{3}$ .

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Step 3.1

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

$- 8 a^{\frac{2}{3} \cdot 3} b^{3} {(a b^{\frac{1}{3}})}^{3}$

Step 3.2

Cancel the common factor of $3$ .

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Step 3.2.1

Cancel the common factor.

$- 8 a^{\frac{2}{3} \cdot 3} b^{3} {(a b^{\frac{1}{3}})}^{3}$

Step 3.2.2

Rewrite the expression.

$- 8 a^{2} b^{3} {(a b^{\frac{1}{3}})}^{3}$

Step 4

Apply the product rule to $a b^{\frac{1}{3}}$ .

$- 8 a^{2} b^{3} (a^{3} {(b^{\frac{1}{3}})}^{3})$

Step 5

Multiply $a^{2}$ by $a^{3}$ by adding the exponents.

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Step 5.1

Move $a^{3}$ .

$- 8 (a^{3} a^{2}) b^{3} {(b^{\frac{1}{3}})}^{3}$

Step 5.2

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

$- 8 a^{3 + 2} b^{3} {(b^{\frac{1}{3}})}^{3}$

Step 5.3

Add $3$ and $2$ .

$- 8 a^{5} b^{3} {(b^{\frac{1}{3}})}^{3}$

Step 6

Multiply the exponents in ${(b^{\frac{1}{3}})}^{3}$ .

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Step 6.1

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

$- 8 a^{5} b^{3} b^{\frac{1}{3} \cdot 3}$

Step 6.2

Cancel the common factor of $3$ .

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Step 6.2.1

Cancel the common factor.

$- 8 a^{5} b^{3} b^{\frac{1}{3} \cdot 3}$

Step 6.2.2

Rewrite the expression.

$- 8 a^{5} b^{3} b^{1}$

Step 7

Multiply $b^{3}$ by $b^{1}$ by adding the exponents.

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Step 7.1

Move $b^{1}$ .

$- 8 a^{5} (b^{1} b^{3})$

Step 7.2

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

$- 8 a^{5} b^{1 + 3}$

Step 7.3

Add $1$ and $3$ .

$- 8 a^{5} b^{4}$