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The expression to simplify is \( 64^{\frac{1}{2}} \times 4^{\frac{3}{2}} \).

Let's break down this expression step-by-step:

1. **Simplify \( 64^{\frac{1}{2}} \):**  
   \( 64^{\frac{1}{2}} \) means the square root of 64.  
   \( \sqrt{64} = 8 \).  
   So, \( 64^{\frac{1}{2}} = 8 \).

2. **Simplify \( 4^{\frac{3}{2}} \):**  
   \( 4^{\frac{3}{2}} \) can be interpreted as \( (4^{\frac{1}{2}})^3 \).  
   First, find \( 4^{\frac{1}{2}} = \sqrt{4} = 2 \).  
   Then raise this to the power of 3: \( 2^3 = 8 \).  
   So, \( 4^{\frac{3}{2}} = 8 \).

3. **Multiply the simplified terms:**  
   Now, multiply the two results:  
   \( 8 \times 8 = 64 \).

Therefore, the simplified result of \( 64^{\frac{1}{2}} \times 4^{\frac{3}{2}} \) is **64**.

Would you like more details or have any questions about this process?

Here are 5 related questions for further exploration:
1. How do you simplify expressions involving fractional exponents?
2. What is the general rule for multiplying terms with fractional exponents?
3. How do fractional exponents relate to roots and powers?
4. How would you simplify \( 81^{\frac{3}{4}} \)?
5. What is the difference between \( a^{\frac{m}{n}} \) and \( (a^m)^{\frac{1}{n}} \)?

**Tip:** When dealing with fractional exponents, remember that \( a^{\frac{m}{n}} \) means the \( n \)-th root of \( a \) raised to the \( m \)-th power.

Learn how to simplify the exponential expression 64^(1/2) * 4^(3/2) with a step-by-step breakdown. This problem involves understanding fractional exponents, roots, and exponentiation rules. Suitable for students in Grades 9-10.

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