Given the rule involving positive numbers $a$ and $b$:

$a^b = (\sqrt{a})^{2b}$

Let's analyze and verify this rule step by step.

Step-by-Step Verification:

1. Expression on the Left: $a^b$

2. Expression on the Right: $(\sqrt{a})^{2b}$

3. Simplify $\sqrt{a}$: Recall that $\sqrt{a} = a^{1/2}$.

4. Substitute $\sqrt{a}$ into the right-hand side expression: $(\sqrt{a})^{2b} = (a^{1/2})^{2b}$

5. Apply the power of a power rule $(x^m)^n = x^{mn}$: $(a^{1/2})^{2b} = a^{(1/2) \cdot 2b}$

6. Simplify the exponent: $a^{(1/2) \cdot 2b} = a^b$

Hence, we have shown that: $a^b = (\sqrt{a})^{2b}$

This verifies the given rule.

Do you have any questions or need further details on any part of this verification? Here are 5 related questions you might want to explore:

1. Can you explain the power of a power rule in more detail?
2. How does this rule change if $a$ and $b$ are not positive numbers?
3. What are some practical applications of this rule?
4. Can you provide an example with specific values for $a$ and $b$?
5. How does this rule relate to logarithms?

Tip: When dealing with exponents, always remember the fundamental properties of exponents, such as the power of a power rule, which can help simplify complex expressions.