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

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

Step 1

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

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

Step 2

Multiply $c^{7}$ by $c^{0}$ by adding the exponents.

Move $c^{0}$ .

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

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

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

Add $0$ and $7$ .

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

Step 3

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

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

Apply the product rule to $- \frac{3 b c^{7}}{2 a}$ .

${(- 1)}^{2} {(\frac{3 b c^{7}}{2 a})}^{2}$

Apply the product rule to $\frac{3 b c^{7}}{2 a}$ .

${(- 1)}^{2} \frac{{(3 b c^{7})}^{2}}{{(2 a)}^{2}}$

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

${(- 1)}^{2} \frac{{(3 b)}^{2} {(c^{7})}^{2}}{{(2 a)}^{2}}$

Apply the product rule to $3 b$ .

${(- 1)}^{2} \frac{3^{2} b^{2} {(c^{7})}^{2}}{{(2 a)}^{2}}$

Apply the product rule to $2 a$ .

${(- 1)}^{2} \frac{3^{2} b^{2} {(c^{7})}^{2}}{2^{2} a^{2}}$

Step 8

Simplify the expression.

Raise $- 1$ to the power of $2$ .

$1 \frac{3^{2} b^{2} {(c^{7})}^{2}}{2^{2} a^{2}}$

Multiply $\frac{3^{2} b^{2} {(c^{7})}^{2}}{2^{2} a^{2}}$ by $1$ .

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

Step 9

Simplify the numerator.

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

$\frac{9 b^{2} {(c^{7})}^{2}}{2^{2} a^{2}}$

Multiply the exponents in ${(c^{7})}^{2}$ .

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

$\frac{9 b^{2} c^{7 \cdot 2}}{2^{2} a^{2}}$

Multiply $7$ by $2$ .

$\frac{9 b^{2} c^{14}}{2^{2} a^{2}}$

Step 10

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

$\frac{9 b^{2} c^{14}}{4 a^{2}}$

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

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