I am doing exponents and surds. How do I simplify 4 to the power of (3 divided by 2)

To simplify \( 4^{\frac{3}{2}} \), follow these steps:

### Step 1: Rewrite the base as a power of 2
Since \( 4 = 2^2 \), we can rewrite the expression:
\[
4^{\frac{3}{2}} = (2^2)^{\frac{3}{2}}
\]

### Step 2: Apply the power of a power rule
Using the rule \( (a^m)^n = a^{m \cdot n} \), simplify the expression:
\[
(2^2)^{\frac{3}{2}} = 2^{2 \cdot \frac{3}{2}} = 2^3
\]

### Step 3: Calculate the result
Now, \( 2^3 = 8 \), so:
\[
4^{\frac{3}{2}} = 8
\]

Thus, \( 4^{\frac{3}{2}} = 8 \).

Would you like more details or have any other questions?

Here are 5 related questions to explore:
1. How do you simplify \( 9^{\frac{3}{2}} \)?
2. What is the meaning of fractional exponents in general?
3. How do you simplify expressions involving both exponents and surds?
4. What is \( 16^{\frac{1}{4}} \)?
5. How do you handle negative fractional exponents?

**Tip**: A fractional exponent like \( a^{\frac{m}{n}} \) means you take the \( n \)-th root of \( a \), then raise the result to the power \( m \).

I am doing exponents and surds. How do I simplify 4 to the power of (3 divided by 2)?

Learn how to simplify 4 raised to the power of 3/2 using the power of a power rule. This detailed step-by-step solution involves rewriting the base as a power of 2 and applying fractional exponent concepts. Ideal for students in Grades 9-11 studying exponents and surds.

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