The ratio `25^(2.5) : 5^(3)` is same as

Options

A. `5 :3`
B. `5 :6`
C. `1 : 25`
D. `25 :1`

Answer

To solve the ratio \( 25^{2.5} : 5^{3} \), we can follow these steps:

### Step 1: Rewrite 25 in terms of base 5
We know that \( 25 = 5^2 \). Therefore, we can rewrite \( 25^{2.5} \) as:
\[ 25^{2.5} = (5^2)^{2.5} \]

### Step 2: Apply the power of a power property
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \), we can simplify:
\[ (5^2)^{2.5} = 5^{2 \cdot 2.5} = 5^{5} \]

### Step 3: Rewrite the ratio
Now we can rewrite the original ratio:
\[ 25^{2.5} : 5^{3} = 5^{5} : 5^{3} \]

### Step 4: Simplify the ratio
Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \), we can simplify the ratio:
\[ \frac{5^{5}}{5^{3}} = 5^{5-3} = 5^{2} \]

### Step 5: Write the final answer
Thus, the ratio \( 25^{2.5} : 5^{3} \) simplifies to:
\[ 5^{2} : 1 \]
or simply \( 25 : 1 \).

### Final Answer:
The ratio \( 25^{2.5} : 5^{3} \) is the same as \( 25 : 1 \).

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