Express 648 in exponential notation.

Options

A. `2^(3)xx3^(3)`
B. `2^(3)xx3^(4)`
C. `2^(2)xx3^(4)`
D. `2^(4)xx3^(3)`

Answer

To express 648 in exponential notation, we need to perform prime factorization. Here’s how we can do it step by step:

### Step 1: Start with the number 648
We begin with the number we want to factor, which is 648.

### Step 2: Divide by the smallest prime number
We start dividing 648 by the smallest prime number, which is 2.

- **648 ÷ 2 = 324**

### Step 3: Continue dividing by 2
Now we take the result (324) and divide it by 2 again.

- **324 ÷ 2 = 162**

### Step 4: Divide by 2 again
We continue with 162 and divide it by 2 once more.

- **162 ÷ 2 = 81**

### Step 5: Move to the next prime number
Since 81 is not divisible by 2, we move to the next prime number, which is 3.

- **81 ÷ 3 = 27**

### Step 6: Continue dividing by 3
Now we take 27 and divide it by 3.

- **27 ÷ 3 = 9**

### Step 7: Divide by 3 again
Next, we divide 9 by 3.

- **9 ÷ 3 = 3**

### Step 8: Divide by 3 one last time
Finally, we divide 3 by 3.

- **3 ÷ 3 = 1**

### Step 9: Collect all the prime factors
Now we can collect all the prime factors we used in the division:

- From our divisions, we have:
- 2 was used 3 times (2 × 2 × 2)
- 3 was used 4 times (3 × 3 × 3 × 3)

### Step 10: Write in exponential notation
Now we can express 648 in exponential notation as follows:

- \( 648 = 2^3 \times 3^4 \)

### Final Answer:
Thus, the exponential notation for 648 is \( 2^3 \times 3^4 \).

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