Evaluate ((-7a^2b^3c^0)/(3a^3b^4c^3))^-4

${(\frac{- 7 a^{2} b^{3} c^{0}}{3 a^{3} b^{4} c^{3}})}^{- 4}$

Step 1

Move $c^{0}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{- 7 a^{2} b^{3}}{3 a^{3} b^{4} c^{3} c^{0}})}^{- 4}$

Step 2

Multiply $c^{3}$ by $c^{0}$ by adding the exponents.

Move $c^{0}$ .

${(\frac{- 7 a^{2} b^{3}}{3 a^{3} b^{4} (c^{0} c^{3})})}^{- 4}$

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

${(\frac{- 7 a^{2} b^{3}}{3 a^{3} b^{4} c^{0 + 3}})}^{- 4}$

Add $0$ and $3$ .

${(\frac{- 7 a^{2} b^{3}}{3 a^{3} b^{4} c^{3}})}^{- 4}$

Step 3

Cancel the common factor of $a^{2}$ and $a^{3}$ .

Factor $a^{2}$ out of $- 7 a^{2} b^{3}$ .

${(\frac{a^{2} (- 7 b^{3})}{3 a^{3} b^{4} c^{3}})}^{- 4}$

Cancel the common factors.

Factor $a^{2}$ out of $3 a^{3} b^{4} c^{3}$ .

${(\frac{a^{2} (- 7 b^{3})}{a^{2} (3 a b^{4} c^{3})})}^{- 4}$

Cancel the common factor.

${(\frac{a^{2} (- 7 b^{3})}{a^{2} (3 a b^{4} c^{3})})}^{- 4}$

Rewrite the expression.

${(\frac{- 7 b^{3}}{3 a b^{4} c^{3}})}^{- 4}$

Step 4

Cancel the common factor of $b^{3}$ and $b^{4}$ .

Factor $b^{3}$ out of $- 7 b^{3}$ .

${(\frac{b^{3} \cdot - 7}{3 a b^{4} c^{3}})}^{- 4}$

Cancel the common factors.

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

${(\frac{b^{3} \cdot - 7}{b^{3} (3 a b c^{3})})}^{- 4}$

Cancel the common factor.

${(\frac{b^{3} \cdot - 7}{b^{3} (3 a b c^{3})})}^{- 4}$

Rewrite the expression.

${(\frac{- 7}{3 a b c^{3}})}^{- 4}$

Step 5

Move the negative in front of the fraction.

${(- \frac{7}{3 a b c^{3}})}^{- 4}$

Step 6

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

${(- \frac{3 a b c^{3}}{7})}^{4}$

Step 7

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

Apply the product rule to $- \frac{3 a b c^{3}}{7}$ .

${(- 1)}^{4} {(\frac{3 a b c^{3}}{7})}^{4}$

Apply the product rule to $\frac{3 a b c^{3}}{7}$ .

${(- 1)}^{4} \frac{{(3 a b c^{3})}^{4}}{7^{4}}$

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

${(- 1)}^{4} \frac{{(3 a b)}^{4} {(c^{3})}^{4}}{7^{4}}$

Apply the product rule to $3 a b$ .

${(- 1)}^{4} \frac{{(3 a)}^{4} b^{4} {(c^{3})}^{4}}{7^{4}}$

Apply the product rule to $3 a$ .

${(- 1)}^{4} \frac{3^{4} a^{4} b^{4} {(c^{3})}^{4}}{7^{4}}$

Step 8

Simplify the expression.

Raise $- 1$ to the power of $4$ .

$1 \frac{3^{4} a^{4} b^{4} {(c^{3})}^{4}}{7^{4}}$

Multiply $\frac{3^{4} a^{4} b^{4} {(c^{3})}^{4}}{7^{4}}$ by $1$ .

$\frac{3^{4} a^{4} b^{4} {(c^{3})}^{4}}{7^{4}}$

Step 9

Simplify the numerator.

Raise $3$ to the power of $4$ .

$\frac{81 a^{4} b^{4} {(c^{3})}^{4}}{7^{4}}$

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

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

$\frac{81 a^{4} b^{4} c^{3 \cdot 4}}{7^{4}}$

Multiply $3$ by $4$ .

$\frac{81 a^{4} b^{4} c^{12}}{7^{4}}$

Step 10

Raise $7$ to the power of $4$ .

$\frac{81 a^{4} b^{4} c^{12}}{2401}$

Evaluate ((-7a^2b^3c^0)/(3a^3b^4c^3))^-4

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