The given expression is:

$\sqrt{4(x-5)^2}$

To simplify this, we analyze the square root and the squared term.

Step 1: Separate the square root

The square root of a product can be expressed as the product of square roots:

$\sqrt{4(x-5)^2} = \sqrt{4} \cdot \sqrt{(x-5)^2}$

Step 2: Simplify each part

• $\sqrt{4} = 2$
• $\sqrt{(x-5)^2} = |x-5|$ (The absolute value ensures the result is non-negative, which is required for square roots.)

Final Answer:

$\sqrt{4(x-5)^2} = 2|x-5|$

This is the simplified form. Let me know if you'd like further clarification or examples!

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5 Related Questions:

1. How do we handle square roots of other quadratic expressions?
2. What is the significance of the absolute value in this solution?
3. How would the solution change if the equation involved a cube root instead?
4. Can you demonstrate solving a similar expression, such as $\sqrt{9(x+3)^2}$?
5. What happens if $x-5$ is explicitly negative in this scenario?

Tip:

Always remember that the square root of a square returns the absolute value of the original expression!