The square root of $20a^3b$ can be simplified by factoring and taking out square terms. Here's the step-by-step process:

Step 1: Factor $20a^3b$

Break each component into square factors:

• $20 = 4 \times 5$ (where $4$ is a perfect square),
• $a^3 = a^2 \cdot a$ (where $a^2$ is a perfect square),
• $b$ remains as is since it isn't a perfect square.

So, we rewrite: $20a^3b = (4 \cdot 5)(a^2 \cdot a)b.$

Step 2: Apply the square root

Using the property $\sqrt{xy} = \sqrt{x} \cdot \sqrt{y}$, take the square root of each term: $\sqrt{20a^3b} = \sqrt{4} \cdot \sqrt{5} \cdot \sqrt{a^2} \cdot \sqrt{a} \cdot \sqrt{b}.$

Step 3: Simplify the terms

• $\sqrt{4} = 2$,
• $\sqrt{5}$ stays as is,
• $\sqrt{a^2} = a$,
• $\sqrt{a}$ stays as is,
• $\sqrt{b}$ stays as is.

Thus: $\sqrt{20a^3b} = 2a\sqrt{5ab}.$

Final Answer:

$\boxed{2a\sqrt{5ab}}$

Would you like additional explanations or examples?

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Here are 5 related questions:

1. How do you simplify the square root of a product involving higher powers, like $\sqrt{45x^5y^3}$?
2. What are the rules for simplifying square roots with variables?
3. How do you handle cube roots for similar expressions?
4. How do you rationalize the denominator when the square root involves a fraction?
5. What happens if $a$ or $b$ is negative in the original expression?

Tip: Always group terms into perfect squares when simplifying square roots; this makes it easier to extract terms.