Precalculus Examples

${(8 a^{6} b^{\frac{3}{2}})}^{\frac{2}{3}}$

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 $8 a^{6} b^{\frac{3}{2}}$ .

${(8 a^{6})}^{\frac{2}{3}} {(b^{\frac{3}{2}})}^{\frac{2}{3}}$

Step 1.2

Apply the product rule to $8 a^{6}$ .

$8^{\frac{2}{3}} {(a^{6})}^{\frac{2}{3}} {(b^{\frac{3}{2}})}^{\frac{2}{3}}$

Step 2

Simplify the expression.

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Step 2.1

Rewrite $8$ as $2^{3}$ .

${(2^{3})}^{\frac{2}{3}} {(a^{6})}^{\frac{2}{3}} {(b^{\frac{3}{2}})}^{\frac{2}{3}}$

Step 2.2

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

$2^{3 (\frac{2}{3})} {(a^{6})}^{\frac{2}{3}} {(b^{\frac{3}{2}})}^{\frac{2}{3}}$

Step 3

Cancel the common factor of $3$ .

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Step 3.1

Cancel the common factor.

$2^{3 (\frac{2}{3})} {(a^{6})}^{\frac{2}{3}} {(b^{\frac{3}{2}})}^{\frac{2}{3}}$

Step 3.2

Rewrite the expression.

$2^{2} {(a^{6})}^{\frac{2}{3}} {(b^{\frac{3}{2}})}^{\frac{2}{3}}$

Step 4

Simplify the expression.

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Step 4.1

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

$4 {(a^{6})}^{\frac{2}{3}} {(b^{\frac{3}{2}})}^{\frac{2}{3}}$

Step 4.2

Multiply the exponents in ${(a^{6})}^{\frac{2}{3}}$ .

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Step 4.2.1

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

$4 a^{6 (\frac{2}{3})} {(b^{\frac{3}{2}})}^{\frac{2}{3}}$

Step 4.2.2

Cancel the common factor of $3$ .

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Step 4.2.2.1

Factor $3$ out of $6$ .

$4 a^{3 (2) \frac{2}{3}} {(b^{\frac{3}{2}})}^{\frac{2}{3}}$

Step 4.2.2.2

Cancel the common factor.

$4 a^{3 \cdot 2 \frac{2}{3}} {(b^{\frac{3}{2}})}^{\frac{2}{3}}$

Step 4.2.2.3

Rewrite the expression.

$4 a^{2 \cdot 2} {(b^{\frac{3}{2}})}^{\frac{2}{3}}$

Step 4.2.3

Multiply $2$ by $2$ .

$4 a^{4} {(b^{\frac{3}{2}})}^{\frac{2}{3}}$

Step 4.3

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

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Step 4.3.1

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

$4 a^{4} b^{\frac{3}{2} \cdot \frac{2}{3}}$

Step 4.3.2

Cancel the common factor of $3$ .

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Step 4.3.2.1

Cancel the common factor.

$4 a^{4} b^{\frac{3}{2} \cdot \frac{2}{3}}$

Step 4.3.2.2

Rewrite the expression.

$4 a^{4} b^{\frac{1}{2} \cdot 2}$

Step 4.3.3

Cancel the common factor of $2$ .

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Step 4.3.3.1

Cancel the common factor.

$4 a^{4} b^{\frac{1}{2} \cdot 2}$

Step 4.3.3.2

Rewrite the expression.

$4 a^{4} b^{1}$

Step 5

Simplify.

$4 a^{4} b$