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To solve the given expression \( \sqrt{z^3} \sqrt[3]{z^8} \) as a single radical using the smallest possible root, follow these steps:

### Step 1: Express the radicals as fractional exponents

- \( \sqrt{z^3} \) is the same as \( z^{3/2} \) because \( \sqrt{x} \) is \( x^{1/2} \).
- \( \sqrt[3]{z^8} \) is the same as \( z^{8/3} \) because \( \sqrt[3]{x} \) is \( x^{1/3} \).

Now we have:
\[
z^{3/2} \times z^{8/3}
\]

### Step 2: Add the exponents

To multiply terms with the same base, we add the exponents:
\[
z^{3/2 + 8/3}
\]
To add these fractions, find a common denominator. The least common denominator (LCD) of 2 and 3 is 6.

Convert each fraction:
\[
\frac{3}{2} = \frac{9}{6}, \quad \frac{8}{3} = \frac{16}{6}
\]
Now, add the fractions:
\[
\frac{9}{6} + \frac{16}{6} = \frac{25}{6}
\]

### Step 3: Express as a single radical

The exponent \( \frac{25}{6} \) can be written as:
\[
z^{25/6} = \sqrt[6]{z^{25}}
\]

Thus, the expression simplified to a single radical is:
\[
\sqrt[6]{z^{25}}
\]

Would you like more details on any of the steps, or do you have any questions?

Here are 5 related questions for practice:
1. How do you simplify \( \sqrt[3]{z^5} \times \sqrt{z^2} \)?
2. What is the process for converting a radical to a fractional exponent?
3. How would you simplify \( \sqrt[4]{z^8} \times \sqrt[2]{z^6} \)?
4. How do you find the least common denominator when adding fractions?
5. Can you explain the difference between multiplying and dividing powers with the same base?

**Tip:** When simplifying expressions involving radicals and exponents, always express them in terms of fractional exponents to simplify operations like multiplication and division.

Write \(\sqrt{z^3} \sqrt[3]{z^8}\) as a single radical using the smallest possible root.

Learn how to simplify the expression \(\sqrt{z^3} \sqrt[3]{z^8}\) as a single radical using the smallest possible root. This problem involves converting radicals to fractional exponents, adding exponents with a common base, and understanding exponent properties. Suitable for students in Grades 9-10.

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