Simplify ((18a^3b^-2)/(21a^-2b^-4))^-3

${(\frac{18 a^{3} b^{- 2}}{21 a^{- 2} b^{- 4}})}^{- 3}$

Step 1

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

${(\frac{18 a^{3} b^{- 2} a^{2}}{21 b^{- 4}})}^{- 3}$

Step 2

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

${(\frac{18 a^{3} b^{- 2} a^{2} b^{4}}{21})}^{- 3}$

Step 3

Simplify the numerator.

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

Move $a^{2}$ .

${(\frac{18 (a^{2} a^{3}) b^{- 2} b^{4}}{21})}^{- 3}$

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

${(\frac{18 a^{2 + 3} b^{- 2} b^{4}}{21})}^{- 3}$

Add $2$ and $3$ .

${(\frac{18 a^{5} b^{- 2} b^{4}}{21})}^{- 3}$

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

Move $b^{4}$ .

${(\frac{18 a^{5} (b^{4} b^{- 2})}{21})}^{- 3}$

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

${(\frac{18 a^{5} b^{4 - 2}}{21})}^{- 3}$

Subtract $2$ from $4$ .

${(\frac{18 a^{5} b^{2}}{21})}^{- 3}$

Step 4

Cancel the common factor of $18$ and $21$ .

Factor $3$ out of $18 a^{5} b^{2}$ .

${(\frac{3 (6 a^{5} b^{2})}{21})}^{- 3}$

Cancel the common factors.

Factor $3$ out of $21$ .

${(\frac{3 (6 a^{5} b^{2})}{3 (7)})}^{- 3}$

Cancel the common factor.

${(\frac{3 (6 a^{5} b^{2})}{3 \cdot 7})}^{- 3}$

Rewrite the expression.

${(\frac{6 a^{5} b^{2}}{7})}^{- 3}$

Step 5

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

${(\frac{7}{6 a^{5} b^{2}})}^{3}$

Step 6

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

Apply the product rule to $\frac{7}{6 a^{5} b^{2}}$ .

$\frac{7^{3}}{{(6 a^{5} b^{2})}^{3}}$

Apply the product rule to $6 a^{5} b^{2}$ .

$\frac{7^{3}}{{(6 a^{5})}^{3} {(b^{2})}^{3}}$

Apply the product rule to $6 a^{5}$ .

$\frac{7^{3}}{6^{3} {(a^{5})}^{3} {(b^{2})}^{3}}$

Step 7

Raise $7$ to the power of $3$ .

$\frac{343}{6^{3} {(a^{5})}^{3} {(b^{2})}^{3}}$

Step 8

Simplify the denominator.

Raise $6$ to the power of $3$ .

$\frac{343}{216 {(a^{5})}^{3} {(b^{2})}^{3}}$

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

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

$\frac{343}{216 a^{5 \cdot 3} {(b^{2})}^{3}}$

Multiply $5$ by $3$ .

$\frac{343}{216 a^{15} {(b^{2})}^{3}}$

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

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

$\frac{343}{216 a^{15} b^{2 \cdot 3}}$

Multiply $2$ by $3$ .

$\frac{343}{216 a^{15} b^{6}}$

Simplify ((18a^3b^-2)/(21a^-2b^-4))^-3

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