Express each of the following rational number in exponential form
` ( -32)/(3125) `

Answer

To express the rational number \(-\frac{32}{3125}\) in exponential form, we will follow these steps:

### Step 1: Factor the numerator and denominator
First, we need to factor both the numerator (-32) and the denominator (3125) into their prime factors.

- **Numerator**:
\(-32\) can be factored as:
\[ -32 = -1 \times 32 = -1 \times 2^5 \]

- **Denominator**:
\(3125\) can be factored as:
\[ 3125 = 5^5 \]
(since \(3125 = 5 \times 625 = 5 \times 5 \times 125 = 5 \times 5 \times 5 \times 25 = 5 \times 5 \times 5 \times 5 \times 5\))

### Step 2: Write the fraction with the factors
Now we can rewrite the fraction using these factors:
\[ -\frac{32}{3125} = -\frac{-1 \times 2^5}{5^5} \]

### Step 3: Simplify the expression
This can be simplified to:
\[ -\frac{2^5}{5^5} = -\left(\frac{2}{5}\right)^5 \]

### Step 4: Write in exponential form
Thus, the final expression in exponential form is:
\[ -\left(\frac{2}{5}\right)^5 \]

### Final Answer:
\[ -\frac{32}{3125} = -\left(\frac{2}{5}\right)^5 \]

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