Express each of the following as a product of prime factors only in exponential form:

270

Answer

To express 270 as a product of prime factors in exponential form, we can follow these steps:

### Step 1: Start with the number
We have the number 270.

### Step 2: Divide by the smallest prime number
We start dividing 270 by the smallest prime number, which is 2.
- 270 ÷ 2 = 135
So, we can write:
\[ 270 = 2 \times 135 \]

### Step 3: Continue factoring 135
Next, we factor 135. The smallest prime number that divides 135 is 3.
- 135 ÷ 3 = 45
So, we can write:
\[ 135 = 3 \times 45 \]
Thus,
\[ 270 = 2 \times 3 \times 45 \]

### Step 4: Factor 45
Now, we factor 45. The smallest prime number that divides 45 is again 3.
- 45 ÷ 3 = 15
So, we can write:
\[ 45 = 3 \times 15 \]
Thus,
\[ 270 = 2 \times 3 \times 3 \times 15 \]

### Step 5: Factor 15
Next, we factor 15. The smallest prime number that divides 15 is 3.
- 15 ÷ 3 = 5
So, we can write:
\[ 15 = 3 \times 5 \]
Thus,
\[ 270 = 2 \times 3 \times 3 \times 3 \times 5 \]

### Step 6: Combine the prime factors
Now we can combine the prime factors:
- We have three 3's, which can be expressed as \(3^3\).
- We have one 2 and one 5.

Thus, we can write:
\[ 270 = 2^1 \times 3^3 \times 5^1 \]

### Final Answer
The final expression of 270 as a product of prime factors in exponential form is:
\[ 270 = 2^1 \times 3^3 \times 5^1 \]

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