https://file.aiquickdraw.com/mathbot/file/1726736242522lo4k5b0w.jpg

Let's go step by step through the problems shown in the image:

### Problem 2:

\[
\frac{\sqrt[6]{64}}{\sqrt{16}} \div \frac{\sqrt[5]{32}}{\sqrt[3]{8}} = ?
\]

**Step 1**: Simplify each radical expression.
- \( \sqrt[6]{64} \): Since \( 64 = 2^6 \), \( \sqrt[6]{64} = 2 \).
- \( \sqrt{16} = 4 \).
- \( \sqrt[5]{32} \): Since \( 32 = 2^5 \), \( \sqrt[5]{32} = 2 \).
- \( \sqrt[3]{8} \): Since \( 8 = 2^3 \), \( \sqrt[3]{8} = 2 \).

So, the expression becomes:

\[
\frac{2}{4} \div \frac{2}{2}
\]

**Step 2**: Perform the division of fractions.
- \( \frac{2}{4} = \frac{1}{2} \),
- \( \frac{2}{2} = 1 \).

Now the expression is:

\[
\frac{1}{2} \div 1 = \frac{1}{2}
\]

Thus, the result of problem 2 is:

\[
\boxed{\frac{1}{2}}
\]

---

### Problem 3:

\[
\frac{1}{32} \times \frac{1}{8} \div \frac{\frac{1}{16}}{\frac{1}{2}} = ?
\]

**Step 1**: Multiply the fractions on the left.
\[
\frac{1}{32} \times \frac{1}{8} = \frac{1 \times 1}{32 \times 8} = \frac{1}{256}
\]

**Step 2**: Simplify the division of fractions.
\[
\frac{\frac{1}{16}}{\frac{1}{2}} = \frac{1}{16} \times \frac{2}{1} = \frac{2}{16} = \frac{1}{8}
\]

**Step 3**: Now divide the two results.
\[
\frac{1}{256} \div \frac{1}{8} = \frac{1}{256} \times \frac{8}{1} = \frac{8}{256} = \frac{1}{32}
\]

Thus, the result of problem 3 is:

\[
\boxed{\frac{1}{32}}
\]

---

Let me know if you need further details or clarifications. Here are 5 related questions you might want to consider:

1. How do you simplify roots of powers of 2 in general?
2. What is the rule for dividing fractions?
3. How can you quickly recognize perfect powers in radical expressions?
4. Why does multiplying fractions result in multiplying both the numerators and denominators?
5. What other strategies can you use to handle radical expressions involving different roots?

**Tip**: When dealing with radicals, try to express the base as a power of a smaller number for easier simplification.

Simplify the following expressions:
1) (6th root of 64) divided by (square root of 16) times (5th root of 32) divided by (cube root of 8)
2) (1/32) times (1/8) divided by (1/16) divided by (1/2)

This tutorial covers simplifying radical expressions, including 6th, 5th, and cube roots. It also demonstrates how to divide and multiply fractions. Suitable for Grades 8-10.

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