Let's simplify the expression step by step:

We have: $\sqrt{ \dfrac{ 20x^5 }{ 6x^2 } }$

Step 1: Simplify the fraction inside the square root

First, simplify the constants and powers of $x$.

• Constants: $\dfrac{20}{6} = \dfrac{10}{3}$
• Powers of $x$: $\dfrac{x^5}{x^2} = x^{5-2} = x^3$

Thus, the expression becomes: $\sqrt{ \dfrac{10x^3}{3} }$

Step 2: Apply the square root

We now apply the square root to the fraction: $\sqrt{\dfrac{10x^3}{3}} = \dfrac{\sqrt{10x^3}}{\sqrt{3}}$

We can simplify the square root of $x^3$: $\sqrt{x^3} = x^{3/2} = x \cdot x^{1/2}$

So, we have: $\dfrac{\sqrt{10} \cdot x^{3/2}}{\sqrt{3}}$

Final Answer:

$\dfrac{\sqrt{10} \cdot x^{3/2}}{\sqrt{3}}$

Would you like further clarification on any part of the process?

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Follow-up Questions:

1. How do you handle square roots with exponents in algebraic expressions?
2. Can you simplify more complicated rational expressions inside a square root?
3. How do you work with fractional exponents in simplifications?
4. What is the role of rationalizing the denominator when simplifying square roots in fractions?
5. How do you deal with expressions involving higher powers of $x$ under a square root?

Tip: When simplifying powers of $x$, always subtract the exponents if dividing like terms, and remember that a square root of a power simplifies to a fractional exponent.