Express each of the following rational number in exponential form
` (256)/(625)`

Answer

To express the rational number \( \frac{256}{625} \) in exponential form, we will follow these steps:

### Step 1: Factor the numerator and denominator
First, we need to factor both 256 and 625 into their prime factors.

- **For 256**:
- \( 256 = 2^8 \) (since \( 256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \))

- **For 625**:
- \( 625 = 5^4 \) (since \( 625 = 5 \times 5 \times 5 \times 5 \))

### Step 2: Write the fraction in terms of its prime factors
Now we can rewrite the fraction using the prime factorization we found:

\[ \frac{256}{625} = \frac{2^8}{5^4} \]

### Step 3: Express in exponential form
We can express this fraction in a single exponential form. Since the bases are different, we cannot combine them directly, but we can write it as:

\[ \frac{2^8}{5^4} = \frac{1}{5^4} \cdot 2^8 \]

### Step 4: Final expression
Thus, the final expression in exponential form is:

\[ \frac{256}{625} = 2^8 \cdot 5^{-4} \]

### Summary
The rational number \( \frac{256}{625} \) in exponential form is \( 2^8 \cdot 5^{-4} \).

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