Express `(-30)^(18)` as a product of exponential forms in four different ways.

Answer

To express \((-30)^{18}\) as a product of exponential forms in four different ways, we can use the properties of exponents. Here’s a step-by-step solution:

### Step 1: Understand the expression
The expression we need to work with is \((-30)^{18}\). We can break this down into its components.

### Step 2: Use the property of exponents
We can use the property of exponents which states that \((xy)^n = x^n \cdot y^n\). We can apply this property to separate \(-30\) into \(-1\) and \(30\).

\[ (-30)^{18} = (-1 \cdot 30)^{18} = (-1)^{18} \cdot (30)^{18} \]

Since \((-1)^{18} = 1\) (because any negative number raised to an even power is positive), we can simplify this to:

\[ (-30)^{18} = 1 \cdot (30)^{18} = (30)^{18} \]

### Step 3: Express \(30\) in different ways
Now, we can express \(30\) in different factorizations:

1. **First way:**
\[ 30 = 3 \cdot 10 \]
Therefore,
\[ (30)^{18} = (3 \cdot 10)^{18} = 3^{18} \cdot 10^{18} \]

2. **Second way:**
\[ 30 = 5 \cdot 6 \]
Therefore,
\[ (30)^{18} = (5 \cdot 6)^{18} = 5^{18} \cdot 6^{18} \]

3. **Third way:**
\[ 30 = 2 \cdot 15 \]
Therefore,
\[ (30)^{18} = (2 \cdot 15)^{18} = 2^{18} \cdot 15^{18} \]

4. **Fourth way:**
\[ 30 = 30 \cdot 1 \]
Therefore,
\[ (30)^{18} = (30 \cdot 1)^{18} = 30^{18} \cdot 1^{18} \]

### Final Result
Thus, we can express \((-30)^{18}\) in four different exponential forms as follows:

1. \(3^{18} \cdot 10^{18}\)
2. \(5^{18} \cdot 6^{18}\)
3. \(2^{18} \cdot 15^{18}\)
4. \(30^{18} \cdot 1^{18}\)