Simplify ( cube root of x^2* sixth root of x^4)^-3

${(\sqrt[3]{x^{2}} \cdot \sqrt[6]{x^{4}})}^{- 3}$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

${(\sqrt[3]{x^{2}} \cdot \sqrt[6]{{(x^{2})}^{2}})}^{- 3}$

Rewrite $\sqrt[6]{{(x^{2})}^{2}}$ as $\sqrt[3]{\sqrt{{(x^{2})}^{2}}}$ .

${(\sqrt[3]{x^{2}} \cdot \sqrt[3]{\sqrt{{(x^{2})}^{2}}})}^{- 3}$

Pull terms out from under the radical, assuming positive real numbers.

${(\sqrt[3]{x^{2}} \cdot \sqrt[3]{x^{2}})}^{- 3}$

Multiply $\sqrt[3]{x^{2}} \sqrt[3]{x^{2}}$ .

Raise $\sqrt[3]{x^{2}}$ to the power of $1$ .

${({\sqrt[3]{x^{2}}}^{1} \sqrt[3]{x^{2}})}^{- 3}$

Raise $\sqrt[3]{x^{2}}$ to the power of $1$ .

${({\sqrt[3]{x^{2}}}^{1} {\sqrt[3]{x^{2}}}^{1})}^{- 3}$

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

${({\sqrt[3]{x^{2}}}^{1 + 1})}^{- 3}$

Add $1$ and $1$ .

${({\sqrt[3]{x^{2}}}^{2})}^{- 3}$

Multiply the exponents in ${({\sqrt[3]{x^{2}}}^{2})}^{- 3}$ .

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

${\sqrt[3]{x^{2}}}^{2 \cdot - 3}$

Multiply $2$ by $- 3$ .

${\sqrt[3]{x^{2}}}^{- 6}$

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

$\frac{1}{{\sqrt[3]{x^{2}}}^{6}}$

Rewrite ${\sqrt[3]{x^{2}}}^{6}$ as $x^{4}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$\frac{1}{{(x^{\frac{2}{3}})}^{6}}$

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

$\frac{1}{x^{\frac{2}{3} \cdot 6}}$

Combine $\frac{2}{3}$ and $6$ .

$\frac{1}{x^{\frac{2 \cdot 6}{3}}}$

Multiply $2$ by $6$ .

$\frac{1}{x^{\frac{12}{3}}}$

Cancel the common factor of $12$ and $3$ .

Factor $3$ out of $12$ .

$\frac{1}{x^{\frac{3 \cdot 4}{3}}}$

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

$\frac{1}{x^{\frac{3 \cdot 4}{3 \cdot 1}}}$

Rewrite the expression.

$\frac{1}{x^{\frac{4}{1}}}$

Divide $4$ by $1$ .

$\frac{1}{x^{4}}$