Simplify Square Root Expression: √(144x¹⁰/9x⁴)

To solve this problem, we will proceed as follows:

• Step 1: Simplify the expression inside the square root.
• Step 2: Apply the square root to both the numerator and the denominator separately.
• Step 3: Simplify the resulting expression.

Let's go through each step:

Step 1: Simplify the fraction $\frac{144x^{10}}{9x^4}$.

In the given expression $\frac{144x^{10}}{9x^4}$, separate the numeric part from the variable part:

$\frac{144}{9} = 16$ and $\frac{x^{10}}{x^4} = x^{10-4} = x^6$.

Thus, the expression simplifies to $16x^6$.

Step 2: Apply the square root property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ to each part.

$\sqrt{\frac{144x^{10}}{9x^4}} = \sqrt{16x^6}$.

Now separate into numeric and variable components: $\sqrt{16}\cdot\sqrt{x^6}$.

$\sqrt{16} = 4$ because $4^2 = 16$.

$\sqrt{x^6} = (x^6)^{1/2} = x^{6/2} = x^3$.

Step 3: Combine the results.

Combine the simplified components: $4x^3$.

Therefore, the solution to the problem is $\boxed{4x^3}$.