The ratio of `25^(2.5) : 5^(3)` is same as.
`25^(2.5):5^(3)` का अनुपात निम्न में किसके समान है?

Options

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

Explanation

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

### Step 1: Rewrite \( 25^{2.5} \) in terms of base 5
We know that \( 25 \) can be expressed as \( 5^2 \). 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: Write the ratio
Now we can express the 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:
\[ 5^{5} : 5^{3} = 5^{5-3} : 1 = 5^{2} : 1 \]

### Step 5: Calculate \( 5^{2} \)
Now we calculate \( 5^{2} \):
\[ 5^{2} = 25 \]

### Step 6: Write the final ratio
Thus, the ratio \( 25^{2.5} : 5^{3} \) simplifies to:
\[ 25 : 1 \]

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

---