https://r2.erweima.ai/mathbot/image/1726767629453b4ox6heo.jpg

In the image, two expressions involving radicals are presented. Let's work through them one by one.

### First Expression:
\[
\sqrt{2} \cdot \sqrt{18} = ?
\]
Using the property of square roots:
\[
\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}
\]
we can simplify:
\[
\sqrt{2} \cdot \sqrt{18} = \sqrt{2 \cdot 18} = \sqrt{36} = 6
\]
So, the first expression simplifies to:
\[
6
\]

### Second Expression:
\[
\sqrt[3]{8} \cdot \sqrt[5]{x^6} = ?
\]
We cannot directly simplify these unlike radicals, but we can express each radical separately.

- \(\sqrt[3]{8} = 2\), because \(2^3 = 8\).
- \(\sqrt[5]{x^6} = x^{6/5}\).

Now we combine these:
\[
\sqrt[3]{8} \cdot \sqrt[5]{x^6} = 2 \cdot x^{6/5}
\]

The final result for the second expression is:
\[
2 \cdot x^{6/5}
\]

Would you like further details on any of these steps?

### Related Questions:
1. How can you combine radicals with different indices?
2. What are the general rules for simplifying radicals involving variables?
3. Can \(\sqrt[5]{x^6}\) be written in a different form using properties of exponents?
4. How do you rationalize the denominator if there are cube roots involved?
5. What is the relationship between cube roots and fractional exponents?

### Tip:
When multiplying radicals with different indices, converting them to fractional exponents can often make the process easier to handle!

√2 · √18 = √36 = 6 and ∛8 · ∜x⁶ = ∛(8x⁶)

This page provides a detailed explanation on how to simplify two expressions involving radicals. The first simplifies √2 · √18 to √36 = 6, and the second expression shows how to handle cube and fourth roots ∛8 · ∜x⁶. Explore key concepts such as radical multiplication and exponent rules.

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