Simplify (3x^-3)^-4x^-4

${(3 x^{- 3})}^{- 4} x^{- 4}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(3 \frac{1}{x^{3}})}^{- 4} x^{- 4}$

Step 2

Combine $3$ and $\frac{1}{x^{3}}$ .

${(\frac{3}{x^{3}})}^{- 4} x^{- 4}$

Step 3

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

${(\frac{x^{3}}{3})}^{4} x^{- 4}$

Step 4

Apply the product rule to $\frac{x^{3}}{3}$ .

$\frac{{(x^{3})}^{4}}{3^{4}} x^{- 4}$

Step 5

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

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

$\frac{x^{3 \cdot 4}}{3^{4}} x^{- 4}$

Multiply $3$ by $4$ .

$\frac{x^{12}}{3^{4}} x^{- 4}$

Step 6

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

$\frac{x^{12}}{81} x^{- 4}$

Step 7

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{x^{12}}{81} \cdot \frac{1}{x^{4}}$

Step 8

Combine.

$\frac{x^{12} \cdot 1}{81 x^{4}}$

Step 9

Cancel the common factor of $x^{12}$ and $x^{4}$ .

Factor $x^{4}$ out of $x^{12} \cdot 1$ .

$\frac{x^{4} (x^{8} \cdot 1)}{81 x^{4}}$

Cancel the common factors.

Factor $x^{4}$ out of $81 x^{4}$ .

$\frac{x^{4} (x^{8} \cdot 1)}{x^{4} \cdot 81}$

Cancel the common factor.

$\frac{x^{4} (x^{8} \cdot 1)}{x^{4} \cdot 81}$

Rewrite the expression.

$\frac{x^{8} \cdot 1}{81}$

Step 10

Multiply $x^{8}$ by $1$ .

$\frac{x^{8}}{81}$