Calculus Examples

${(\frac{3 a b^{- 2}}{5 b})}^{3}$

Step 1

Move $b^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{3 a}{5 b \cdot b^{2}})}^{3}$

Step 2

Multiply $b$ by $b^{2}$ by adding the exponents.

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Step 2.1

Move $b^{2}$ .

${(\frac{3 a}{5 (b^{2} b)})}^{3}$

Step 2.2

Multiply $b^{2}$ by $b$ .

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Step 2.2.1

Raise $b$ to the power of $1$ .

${(\frac{3 a}{5 (b^{2} b^{1})})}^{3}$

Step 2.2.2

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

${(\frac{3 a}{5 b^{2 + 1}})}^{3}$

Step 2.3

Add $2$ and $1$ .

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

Step 3

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

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Step 3.1

Apply the product rule to $\frac{3 a}{5 b^{3}}$ .

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

Step 3.2

Apply the product rule to $3 a$ .

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

Step 3.3

Apply the product rule to $5 b^{3}$ .

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

Step 4

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

$\frac{27 a^{3}}{5^{3} {(b^{3})}^{3}}$

Step 5

Simplify the denominator.

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Step 5.1

Raise $5$ to the power of $3$ .

$\frac{27 a^{3}}{125 {(b^{3})}^{3}}$

Step 5.2

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

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Step 5.2.1

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

$\frac{27 a^{3}}{125 b^{3 \cdot 3}}$

Step 5.2.2

Multiply $3$ by $3$ .

$\frac{27 a^{3}}{125 b^{9}}$