Algebra Examples

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

Step 1

Move $b^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2

Simplify the numerator.

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Step 2.1

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

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Step 2.1.1

Move $b^{3}$ .

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

Step 2.1.2

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

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

Step 2.1.3

Subtract $2$ from $3$ .

${(\frac{3 a^{4} b^{1}}{6 a})}^{- 2}$

Step 2.2

Simplify $3 a^{4} b^{1}$ .

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

Step 3

Cancel the common factor of $3$ and $6$ .

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Step 3.1

Factor $3$ out of $3 a^{4} b$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $3$ out of $6 a$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

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

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Step 4.1

Factor $a$ out of $a^{4} b$ .

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $a$ out of $2 a$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Multiply $3$ by $2$ .

$\frac{4}{a^{6} b^{2}}$