Rule:
Positive numbers a, b
a^b = (√a)^(2*b)

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.

Explore and verify the mathematical rule a^b = (√a)^(2b) with a step-by-step explanation. This problem involves exponents and roots, demonstrating the power of a power rule in algebra. Suitable for advanced high school students and beyond.

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