https://file.aiquickdraw.com/mathbot/file/1725978847453x9q0it45.jpg

The expression in the image is:

\[
\frac{\sqrt[4]{k^9}}{\sqrt[5]{k^7}}
\]

We are tasked with expressing this as a single radical using the smallest possible root. Let’s break it down:

### Step-by-Step Solution:

1. **Rewrite the radicals as fractional exponents:**
   - \( \sqrt[4]{k^9} = k^{9/4} \)
   - \( \sqrt[5]{k^7} = k^{7/5} \)

2. **Apply the division rule for exponents:**
   When dividing like bases, we subtract the exponents:
   \[
   \frac{k^{9/4}}{k^{7/5}} = k^{\left(\frac{9}{4} - \frac{7}{5}\right)}
   \]

3. **Find a common denominator:**
   The least common denominator (LCD) between 4 and 5 is 20, so we convert the fractions:
   \[
   \frac{9}{4} = \frac{45}{20}, \quad \frac{7}{5} = \frac{28}{20}
   \]

4. **Subtract the exponents:**
   \[
   \frac{45}{20} - \frac{28}{20} = \frac{17}{20}
   \]

5. **Write the result as a radical:**
   The expression simplifies to:
   \[
   k^{17/20}
   \]
   Which can be written as:
   \[
   \sqrt[20]{k^{17}}
   \]

Thus, the expression \(\frac{\sqrt[4]{k^9}}{\sqrt[5]{k^7}}\) simplifies to \(\sqrt[20]{k^{17}}\).

---

Would you like further details on any of these steps?

Here are 5 related questions for further exploration:
1. How do you simplify an expression with multiple radicals?
2. What is the rule for combining exponents in division?
3. What is the importance of using the smallest root in radical simplifications?
4. How do you find the least common denominator when subtracting fractions?
5. How do fractional exponents relate to radicals?

**Tip:** When working with roots and exponents, it's useful to first convert everything into fractional exponents, making operations like multiplication and division much easier!

Write \(\frac{\sqrt[4]{k^9}}{\sqrt[5]{k^7}}\) as a single radical using the smallest possible root.

Learn how to simplify the expression \(\frac{\sqrt[4]{k^9}}{\sqrt[5]{k^7}}\) into a single radical with the smallest possible root. This solution covers radicals, fractional exponents, and exponent rules. Suitable for high school students in Grades 9-12.

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