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

Video Solution

Solution Steps

00:00 Solve the following problem

00:03 A "regular" root is of the order 2

00:11 When we have a number (A) to the power of (B) in a root of order (C)

00:15 The result equals the number (A) to the power of their product (B times C)

00:19 We'll apply this formula to our exercise

00:25 When we have a root of the multiplication (A times B)

00:28 It can be written as multiplication of the root of each term

00:33 We'll apply this formula to our exercise, and proceed to break down the root

00:44 When we have a number (A) to the power of (B) in a root of order (C)

00:47 The result equals the number (A) to the power of their quotient (B divided by C)

00:52 We'll apply this formula to our exercise, and proceed to calculate the power quotient

01:05 We'll apply this formula again in the reverse direction, converting from power to root

01:18 This is the solution

Step-by-Step Solution

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.