Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Multiply $2$ by $- 3$ .

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

Step 2

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

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

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Multiply $3$ by $2$ .

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

Step 4

Cancel the common factor of $a^{6}$ .

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Step 4.1

Factor $a^{6}$ out of $a^{6} b^{2}$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

$b^{2} {(b^{3})}^{4}$

Step 5

Multiply the exponents in ${(b^{3})}^{4}$ .

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Step 5.1

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

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

Step 5.2

Multiply $3$ by $4$ .

$b^{2} b^{12}$

Step 6

Multiply $b^{2}$ by $b^{12}$ by adding the exponents.

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Step 6.1

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

$b^{2 + 12}$

Step 6.2

Add $2$ and $12$ .

$b^{14}$