`4^(3.5) : 2^(5)` is the same as:

Options

A. `4:1`
B. `2:1`
C. `7:5`
D. `7:10`

Answer

To solve the expression \( \frac{4^{3.5}}{2^5} \), we can follow these steps:

### Step 1: Rewrite the base
First, we can express \( 4 \) in terms of \( 2 \):
\[ 4 = 2^2 \]
So, we can rewrite \( 4^{3.5} \) as:
\[ 4^{3.5} = (2^2)^{3.5} \]

### Step 2: Apply the power of a power property
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \), we can simplify:
\[ (2^2)^{3.5} = 2^{2 \cdot 3.5} = 2^{7} \]

### Step 3: Substitute back into the expression
Now, we can substitute this back into our original expression:
\[ \frac{4^{3.5}}{2^5} = \frac{2^7}{2^5} \]

### Step 4: Apply the quotient of powers property
Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \), we can simplify:
\[ \frac{2^7}{2^5} = 2^{7-5} = 2^{2} \]

### Step 5: Calculate the final result
Now we can calculate \( 2^2 \):
\[ 2^2 = 4 \]

### Step 6: Write the final answer in ratio form
Since the question asks for the ratio, we can express \( 4 \) as:
\[ \frac{4}{1} \]

Thus, the final answer is:
\[ 4 : 1 \]