Simplify ((((4x^-1)^(1/2))^(-2/3))^-1)^3

${({({({(4 x^{- 1})}^{\frac{1}{2}})}^{- \frac{2}{3}})}^{- 1})}^{3}$

Step 1

Multiply the exponents in ${({({({(4 x^{- 1})}^{\frac{1}{2}})}^{- \frac{2}{3}})}^{- 1})}^{3}$ .

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

${({({(4 x^{- 1})}^{\frac{1}{2}})}^{- \frac{2}{3}})}^{- 1 \cdot 3}$

Multiply $- 1$ by $3$ .

${({({(4 x^{- 1})}^{\frac{1}{2}})}^{- \frac{2}{3}})}^{- 3}$

Step 2

Multiply the exponents in ${({({(4 x^{- 1})}^{\frac{1}{2}})}^{- \frac{2}{3}})}^{- 3}$ .

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

${({(4 x^{- 1})}^{\frac{1}{2}})}^{- \frac{2}{3} \cdot - 3}$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{2}{3}$ into the numerator.

${({(4 x^{- 1})}^{\frac{1}{2}})}^{\frac{- 2}{3} \cdot - 3}$

Factor $3$ out of $- 3$ .

${({(4 x^{- 1})}^{\frac{1}{2}})}^{\frac{- 2}{3} \cdot (3 (- 1))}$

Cancel the common factor.

${({(4 x^{- 1})}^{\frac{1}{2}})}^{\frac{- 2}{3} \cdot (3 \cdot - 1)}$

Rewrite the expression.

${({(4 x^{- 1})}^{\frac{1}{2}})}^{- 2 \cdot - 1}$

Multiply $- 2$ by $- 1$ .

${({(4 x^{- 1})}^{\frac{1}{2}})}^{2}$

Step 3

Multiply the exponents in ${({(4 x^{- 1})}^{\frac{1}{2}})}^{2}$ .

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

${(4 x^{- 1})}^{\frac{1}{2} \cdot 2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 x^{- 1})}^{\frac{1}{2} \cdot 2}$

Rewrite the expression.

${(4 x^{- 1})}^{1}$

Step 4

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

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

Step 5

Combine $4$ and $\frac{1}{x}$ .

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

Step 6

Simplify.

$\frac{4}{x}$