Precalculus Examples

${(3 A B)}^{- 1} {(A^{2} B^{- 1})}^{2}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3 A B} {(A^{2} B^{- 1})}^{2}$

Step 2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3 A B} {(A^{2} \frac{1}{B})}^{2}$

Step 3

Combine $A^{2}$ and $\frac{1}{B}$ .

$\frac{1}{3 A B} {(\frac{A^{2}}{B})}^{2}$

Step 4

Apply the product rule to $\frac{A^{2}}{B}$ .

$\frac{1}{3 A B} \cdot \frac{{(A^{2})}^{2}}{B^{2}}$

Step 5

Combine.

$\frac{1 {(A^{2})}^{2}}{3 A B \cdot B^{2}}$

Step 6

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

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Step 6.1

Move $B^{2}$ .

$\frac{1 {(A^{2})}^{2}}{3 A (B^{2} B)}$

Step 6.2

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

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Step 6.2.1

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

$\frac{1 {(A^{2})}^{2}}{3 A (B^{2} B^{1})}$

Step 6.2.2

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

$\frac{1 {(A^{2})}^{2}}{3 A B^{2 + 1}}$

Step 6.3

Add $2$ and $1$ .

$\frac{1 {(A^{2})}^{2}}{3 A B^{3}}$

Step 7

Simplify the expression.

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Step 7.1

Multiply ${(A^{2})}^{2}$ by $1$ .

$\frac{{(A^{2})}^{2}}{3 A B^{3}}$

Step 7.2

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

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Step 7.2.1

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

$\frac{A^{2 \cdot 2}}{3 A B^{3}}$

Step 7.2.2

Multiply $2$ by $2$ .

$\frac{A^{4}}{3 A B^{3}}$

Step 8

Cancel the common factor of $A^{4}$ and $A$ .

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Step 8.1

Factor $A$ out of $A^{4}$ .

$\frac{A \cdot A^{3}}{3 A B^{3}}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $A$ out of $3 A B^{3}$ .

$\frac{A \cdot A^{3}}{A (3 B^{3})}$

Step 8.2.2

Cancel the common factor.

$\frac{A \cdot A^{3}}{A (3 B^{3})}$

Step 8.2.3

Rewrite the expression.

$\frac{A^{3}}{3 B^{3}}$