Simplifying Nested Square Roots: $ \sqrt{\sqrt{256x^4}} $

Math Problem Statement

$\sqrt{\sqrt{256x^4}}$

To solve $\sqrt{\sqrt{256x^4}}$ , we'll break it down step by step:

We first focus on simplifying $\sqrt{256x^4}$ .

$\sqrt{256x^4} = \sqrt{256} \times \sqrt{x^4}$

We know that:

\quad \text{and} \quad \sqrt{x^4} = x^2$$ Thus, we get: $$\sqrt{256x^4} = 16x^2$$ ### Step 2: Apply the outer square root Now, we need to simplify $$\sqrt{16x^2}$$. $$\sqrt{16x^2} = \sqrt{16} \times \sqrt{x^2}$$ Since: $$\sqrt{16} = 4 \quad \text{and} \quad \sqrt{x^2} = |x|$$ Thus, the final result is: $$\sqrt{\sqrt{256x^4}} = 4|x|$$ The absolute value is needed because $$\sqrt{x^2}$$ gives the positive root. ### Final Answer: $$\boxed{4|x|}$$ Would you like more details on this process, or do you have any questions? Here are some related questions you might find interesting: 1. How do you simplify $$\sqrt{x^6}$$? 2. What is the general rule for simplifying nested square roots? 3. Can you simplify $$\sqrt{81x^8}$$? 4. How does the absolute value affect the simplification of square roots? 5. What happens if $$x$$ is negative in the expression $$\sqrt{x^2}$$? **Tip:** Always remember that the square root of a variable squared, $$\sqrt{x^2}$$, equals $$|x|$$, not just $$x$$. This accounts for both positive and negative values of $$x$$.

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Exponents
Square Roots
Absolute Value

$ \sqrt{a^2} = |a| $
$ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} $

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