To express \( 400^{\frac{1}{4}} \) in its simplest radical form:

1. We start by recognizing that \( 400^{\frac{1}{4}} \) represents the fourth root of 400, or \( \sqrt[4]{400} \).

2. Factorize 400:
\[
400 = 2^4 \times 5^2
\]

3. Now, apply the fourth root to each factor:
\[
\sqrt[4]{400} = \sqrt[4]{2^4 \times 5^2} = \sqrt[4]{2^4} \times \sqrt[4]{5^2}
\]

4. Simplifying:
\[
\sqrt[4]{2^4} = 2
\]
\[
\sqrt[4]{5^2} = 5^{\frac{1}{2}} = \sqrt{5}
\]

5. Therefore, the expression becomes:
\[
400^{\frac{1}{4}} = 2\sqrt{5}
\]

So, \( 400^{\frac{1}{4}} \) in simplest radical form is \( 2\sqrt{5} \).

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Would you like any further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How would you express \( 625^{\frac{1}{4}} \) in simplest radical form?
2. What is \( 16^{\frac{1}{4}} \) in radical form?
3. Can you simplify \( 81^{\frac{1}{4}} \)?
4. How would you express \( 500^{\frac{1}{4}} \) in simplest form?
5. How do you simplify \( 25^{\frac{3}{4}} \)?

**Tip:** When simplifying radicals with fractional exponents, break the base into prime factors to apply the root step by step.

Discover how to express 400^(1/4) in simplest radical form with detailed steps. Learn about exponents, radicals, and prime factorization. The solution is perfect for students in Grades 8-10 looking to deepen their understanding of radical expressions.

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