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

Options

A. `4:1`
B. `2:1`
C. `1:2`
D. `1:4`

Answer

To solve the problem of finding the ratio \( 4^{3.5} : 2^{5} \), we can follow these steps:

### Step 1: Rewrite the base 4 in terms of base 2
We know that \( 4 = 2^2 \). Therefore, 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: Rewrite the ratio
Now we can rewrite the ratio \( 4^{3.5} : 2^{5} \) as:
\[ 2^{7} : 2^{5} \]

### Step 4: Use the property of exponents for division
Using the property \( a^m : a^n = a^{m-n} \):
\[ 2^{7} : 2^{5} = 2^{7-5} = 2^{2} \]

### Step 5: Write the final ratio
Now we can express \( 2^{2} \) as:
\[ 2^{2} = 4 \]
Thus, the ratio \( 2^{7} : 2^{5} \) simplifies to:
\[ 4 : 1 \]

### Final Answer
The ratio \( 4^{3.5} : 2^{5} \) is the same as \( 4 : 1 \).

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