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.