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We are given the expression \( \sqrt{\sqrt{k^3}} \) and asked to rewrite it as a single radical using the smallest possible root.

### Step-by-Step Solution:

1. The expression \( \sqrt{\sqrt{k^3}} \) can be rewritten in terms of fractional exponents. Recall that the square root of a number is equivalent to raising that number to the power of \( \frac{1}{2} \). So,
   \[
   \sqrt{k^3} = (k^3)^{\frac{1}{2}} = k^{\frac{3}{2}}.
   \]

2. Now, applying the square root again to \( k^{\frac{3}{2}} \):
   \[
   \sqrt{k^{\frac{3}{2}}} = (k^{\frac{3}{2}})^{\frac{1}{2}} = k^{\frac{3}{4}}.
   \]

3. The result is \( k^{\frac{3}{4}} \), which can also be written as:
   \[
   \sqrt[4]{k^3}.
   \]

Thus, the expression \( \sqrt{\sqrt{k^3}} \) simplifies to \( \sqrt[4]{k^3} \).

### Final Answer:
\[
\sqrt{\sqrt{k^3}} = \sqrt[4]{k^3}.
\]

Would you like more details or have any questions?

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Here are 5 related questions for further exploration:

1. How do you simplify \( \sqrt[3]{\sqrt{k^6}} \) into a single radical?
2. What is the general rule for simplifying nested radicals?
3. How can fractional exponents help simplify radical expressions?
4. Can you express \( \sqrt[4]{k^8} \) as a simpler radical or integer?
5. What happens when you simplify \( \sqrt[3]{k^9} \)?

**Tip:** When dealing with radicals, converting them into fractional exponents often makes simplification much easier!

Write √(√(k^3)) as a single radical using the smallest possible root.

Learn how to simplify √(√(k^3)) as a single radical using the smallest possible root. This problem covers key algebraic concepts, including the simplification of radicals and the application of exponent laws. Suitable for students in Grades 10-12.

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