Calculus Examples

$\frac{{(a^{m} \cdot a^{- n})}^{- 3}}{{(a^{m + n})}^{3}}$

Step 1

Move ${(a^{m} \cdot a^{- n})}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(a^{m + n})}^{3} {(a^{m} \cdot a^{- n})}^{3}}$

Step 2

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

$\frac{1}{{(a^{m + n})}^{3} {(a^{m - n})}^{3}}$

Step 3

Simplify the denominator.

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Step 3.1

Multiply the exponents in ${(a^{m + n})}^{3}$ .

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Step 3.1.1

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

$\frac{1}{a^{(m + n) \cdot 3} {(a^{m - n})}^{3}}$

Step 3.1.2

Apply the distributive property.

$\frac{1}{a^{m \cdot 3 + n \cdot 3} {(a^{m - n})}^{3}}$

Step 3.1.3

Move $3$ to the left of $m$ .

$\frac{1}{a^{3 \cdot m + n \cdot 3} {(a^{m - n})}^{3}}$

Step 3.1.4

Move $3$ to the left of $n$ .

$\frac{1}{a^{3 \cdot m + 3 \cdot n} {(a^{m - n})}^{3}}$

Step 3.1.5

Multiply $3$ by $n$ .

$\frac{1}{a^{3 m + 3 n} {(a^{m - n})}^{3}}$

Step 3.2

Multiply the exponents in ${(a^{m - n})}^{3}$ .

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Step 3.2.1

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

$\frac{1}{a^{3 m + 3 n} a^{(m - n) \cdot 3}}$

Step 3.2.2

Apply the distributive property.

$\frac{1}{a^{3 m + 3 n} a^{m \cdot 3 - n \cdot 3}}$

Step 3.2.3

Move $3$ to the left of $m$ .

$\frac{1}{a^{3 m + 3 n} a^{3 \cdot m - n \cdot 3}}$

Step 3.2.4

Multiply $3$ by $- 1$ .

$\frac{1}{a^{3 m + 3 n} a^{3 m - 3 n}}$

Step 4

Multiply $a^{3 m + 3 n}$ by $a^{3 m - 3 n}$ by adding the exponents.

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Step 4.1

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

$\frac{1}{a^{3 m + 3 n + 3 m - 3 n}}$

Step 4.2

Combine the opposite terms in $3 m + 3 n + 3 m - 3 n$ .

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Step 4.2.1

Subtract $3 n$ from $3 n$ .

$\frac{1}{a^{3 m + 3 m + 0}}$

Step 4.2.2

Add $3 m + 3 m$ and $0$ .

$\frac{1}{a^{3 m + 3 m}}$

Step 4.3

Add $3 m$ and $3 m$ .

$\frac{1}{a^{6 m}}$