Algebra Examples

${(\frac{18 a^{3} b^{- 2}}{27 a^{- 2} b^{- 4}})}^{- 3}$

Step 1

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

${(\frac{18 a^{3} b^{- 2} a^{2}}{27 b^{- 4}})}^{- 3}$

Step 2

Move $b^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{18 a^{3} b^{- 2} a^{2} b^{4}}{27})}^{- 3}$

Step 3

Simplify the numerator.

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Step 3.1

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

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Step 3.1.1

Move $a^{2}$ .

${(\frac{18 (a^{2} a^{3}) b^{- 2} b^{4}}{27})}^{- 3}$

Step 3.1.2

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

${(\frac{18 a^{2 + 3} b^{- 2} b^{4}}{27})}^{- 3}$

Step 3.1.3

Add $2$ and $3$ .

${(\frac{18 a^{5} b^{- 2} b^{4}}{27})}^{- 3}$

Step 3.2

Multiply $b^{- 2}$ by $b^{4}$ by adding the exponents.

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Step 3.2.1

Move $b^{4}$ .

${(\frac{18 a^{5} (b^{4} b^{- 2})}{27})}^{- 3}$

Step 3.2.2

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

${(\frac{18 a^{5} b^{4 - 2}}{27})}^{- 3}$

Step 3.2.3

Subtract $2$ from $4$ .

${(\frac{18 a^{5} b^{2}}{27})}^{- 3}$

Step 4

Cancel the common factor of $18$ and $27$ .

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Step 4.1

Factor $9$ out of $18 a^{5} b^{2}$ .

${(\frac{9 (2 a^{5} b^{2})}{27})}^{- 3}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $9$ out of $27$ .

${(\frac{9 (2 a^{5} b^{2})}{9 (3)})}^{- 3}$

Step 4.2.2

Cancel the common factor.

${(\frac{9 (2 a^{5} b^{2})}{9 \cdot 3})}^{- 3}$

Step 4.2.3

Rewrite the expression.

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

Step 5

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

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

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 $\frac{3}{2 a^{5} b^{2}}$ .

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

Step 6.2

Apply the product rule to $2 a^{5} b^{2}$ .

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

Step 6.3

Apply the product rule to $2 a^{5}$ .

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

Step 7

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

$\frac{27}{2^{3} {(a^{5})}^{3} {(b^{2})}^{3}}$

Step 8

Simplify the denominator.

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Step 8.1

Raise $2$ to the power of $3$ .

$\frac{27}{8 {(a^{5})}^{3} {(b^{2})}^{3}}$

Step 8.2

Multiply the exponents in ${(a^{5})}^{3}$ .

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Step 8.2.1

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

$\frac{27}{8 a^{5 \cdot 3} {(b^{2})}^{3}}$

Step 8.2.2

Multiply $5$ by $3$ .

$\frac{27}{8 a^{15} {(b^{2})}^{3}}$

Step 8.3

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

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Step 8.3.1

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

$\frac{27}{8 a^{15} b^{2 \cdot 3}}$

Step 8.3.2

Multiply $2$ by $3$ .

$\frac{27}{8 a^{15} b^{6}}$