Simplify the Nested Radical: Cube Root of Square Root of 144x³

Complete the following exercise:

$\sqrt[3]{\sqrt{144x^3}}=$

To solve the expression $\sqrt[3]{\sqrt{144x^3}}$, let's proceed step by step:

• Step 1: Express the inner square root using fractional exponents:
  $\sqrt{144x^3} = (144x^3)^{1/2}$
• Step 2: Apply the cube root to the expression obtained in Step 1:
  $\sqrt[3]{(144x^3)^{1/2}} = ((144x^3)^{1/2})^{1/3}$
• Step 3: Use the exponent rule $(a^m)^n = a^{m \cdot n}$:
  $((144x^3)^{1/2})^{1/3} = (144x^3)^{(1/2) \cdot (1/3)} = (144x^3)^{1/6}$
• Step 4: Separate the powers for 144 and $x^3$:
  $(144)^{1/6} \cdot (x^3)^{1/6} = \sqrt[6]{144} \cdot x^{3/6} = \sqrt[6]{144} \cdot x^{1/2} = \sqrt[6]{144} \cdot \sqrt{x}$

Therefore, the simplified expression is $\sqrt[6]{144} \cdot \sqrt{x}$, which corresponds to the third choice.