Solve for X: Square Root of 144 Equals Nested Cube and Fifth Roots

To solve this equation, we will follow these steps:

• Step 1: Simplify the left side of the equation $\sqrt{144}$.
• Step 2: Simplify the right side of the equation $\sqrt[3]{\sqrt[5]{x^{30}}}$.
• Step 3: Equate the simplified expressions and solve for $x$.

Let us go through these steps:

Step 1: Simplify the left side:

The left side of the equation is $\sqrt{144}$, which simplifies to $12$, because $\sqrt{144} = 12$.

Step 2: Simplify the right side:

The expression on the right is $\sqrt[3]{\sqrt[5]{x^{30}}}$. Let's simplify it step by step:

• First, simplify $\sqrt[5]{x^{30}}$:
  - Using the rule $\sqrt[n]{a^m} = a^{m/n}$, we have $\sqrt[5]{x^{30}} = x^{30/5} = x^6$.
• Next, simplify $\sqrt[3]{x^6}$:
  - Again using the same rule, $\sqrt[3]{x^6} = x^{6/3} = x^2$.

Step 3: Equate and solve:

From the previous steps, we get:

$12 = x^2$

Therefore, the solution to the equation is:
$x^2 = 12$.