Simplify (((ab^2c)^-3)/(b^-3))^3

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

Step 1

Simplify the expression.

Move ${(a b^{2} c)}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

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

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

Step 2

Simplify the denominator.

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

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

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

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

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

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

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

Multiply $2$ by $3$ .

${(\frac{b^{3}}{a^{3} b^{6} c^{3}})}^{3}$

Step 3

Cancel the common factor of $b^{3}$ and $b^{6}$ .

Multiply by $1$ .

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

Cancel the common factors.

Factor $b^{3}$ out of $a^{3} b^{6} c^{3}$ .

${(\frac{b^{3} \cdot 1}{b^{3} (a^{3} b^{3} c^{3})})}^{3}$

Cancel the common factor.

${(\frac{b^{3} \cdot 1}{b^{3} (a^{3} b^{3} c^{3})})}^{3}$

Rewrite the expression.

${(\frac{1}{a^{3} b^{3} c^{3}})}^{3}$

Step 4

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

Apply the product rule to $\frac{1}{a^{3} b^{3} c^{3}}$ .

$\frac{1^{3}}{{(a^{3} b^{3} c^{3})}^{3}}$

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

$\frac{1^{3}}{{(a^{3} b^{3})}^{3} {(c^{3})}^{3}}$

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

$\frac{1^{3}}{{(a^{3})}^{3} {(b^{3})}^{3} {(c^{3})}^{3}}$

Step 5

One to any power is one.

$\frac{1}{{(a^{3})}^{3} {(b^{3})}^{3} {(c^{3})}^{3}}$

Step 6

Simplify the denominator.

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

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

$\frac{1}{a^{3 \cdot 3} {(b^{3})}^{3} {(c^{3})}^{3}}$

Multiply $3$ by $3$ .

$\frac{1}{a^{9} {(b^{3})}^{3} {(c^{3})}^{3}}$

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

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

$\frac{1}{a^{9} b^{3 \cdot 3} {(c^{3})}^{3}}$

Multiply $3$ by $3$ .

$\frac{1}{a^{9} b^{9} {(c^{3})}^{3}}$

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

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

$\frac{1}{a^{9} b^{9} c^{3 \cdot 3}}$

Multiply $3$ by $3$ .

$\frac{1}{a^{9} b^{9} c^{9}}$

Simplify (((ab^2c)^-3)/(b^-3))^3

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