simplify: `16^(5/2) div 16^(1/2) =`

Options

A. 250
B. 256
C. 255
D. 200

Answer

To simplify the expression \( 16^{5/2} \div 16^{1/2} \), we can follow these steps:

### Step 1: Rewrite the base
First, we recognize that \( 16 \) can be expressed as a power of \( 2 \):
\[ 16 = 2^4 \]
Thus, we can rewrite the expression:
\[ 16^{5/2} = (2^4)^{5/2} \quad \text{and} \quad 16^{1/2} = (2^4)^{1/2} \]

### Step 2: Apply the power of a power property
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \), we can simplify both parts:
\[ (2^4)^{5/2} = 2^{4 \cdot (5/2)} = 2^{10} \]
\[ (2^4)^{1/2} = 2^{4 \cdot (1/2)} = 2^{2} \]

### Step 3: Rewrite the division
Now we can rewrite the original expression using our new forms:
\[ 16^{5/2} \div 16^{1/2} = 2^{10} \div 2^{2} \]

### Step 4: Apply the quotient of powers property
Using the property of exponents \( \frac{a^n}{a^m} = a^{n-m} \), we can simplify further:
\[ 2^{10} \div 2^{2} = 2^{10 - 2} = 2^{8} \]

### Step 5: Calculate the final value
Now we need to find the value of \( 2^8 \):
\[ 2^8 = 256 \]

### Final Answer
Thus, the simplified result of \( 16^{5/2} \div 16^{1/2} \) is:
\[ \boxed{256} \]

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