To simplify $\sqrt[3]{10} \times \sqrt{5}$, follow these steps:

Step 1: Combine the expressions under the cube root and square root.

You have $\sqrt[3]{10} \times \sqrt{5}$, which involves a cube root and a square root. You can express both terms as exponents:

• $\sqrt[3]{10} = 10^{1/3}$
• $\sqrt{5} = 5^{1/2}$

Thus, the expression becomes: $10^{1/3} \times 5^{1/2}$

Step 2: Keep the terms separate.

Since these two terms have different bases, we cannot directly combine them into one root or exponent without further manipulation. Therefore, we leave the expression as: $10^{1/3} \times 5^{1/2}$

Step 3: Approximate if needed.

If you're looking for an approximation, you can evaluate the expression numerically. Here's how:

$10^{1/3} \approx 2.154 \quad \text{and} \quad 5^{1/2} \approx 2.236$

Multiplying these gives: $2.154 \times 2.236 \approx 4.81$

Final Answer:

The simplified form of the expression remains $10^{1/3} \times 5^{1/2}$, or approximately 4.81.

Let me know if you'd like more details or have any other questions!

Here are some related questions you could explore:

1. How do you simplify the product of cube roots and square roots in general?
2. What is the method to combine different radical terms?
3. How do you approximate cube roots and square roots manually?
4. How do exponents and radicals relate to each other in simplifying expressions?
5. What are the properties of exponents when combining powers with different bases?

Tip: To combine radicals with different indices, express them using exponents, then simplify or approximate as needed!