How can the whole number 378 be expressed as a product of prime numbers with exponents?(1 point)%0D%0AResponses%0D%0A%0D%0A2⋅33⋅7%0D%0A2 times 3 cubed times 7%0D%0A%0D%0A2⋅3⋅3⋅3⋅7%0D%0A2 times 3 times 3 times 3 times 7%0D%0A%0D%0A2⋅27⋅7%0D%0A2 times 27 times 7%0D%0A%0D%0A14⋅27
To express the whole number 378 as a product of prime numbers with exponents, we need to factorize it. Here's how we can do it step-by-step:
1. Begin by dividing 378 by the smallest prime number, which is 2. You will find that 378 divided by 2 is equal to 189, with no remainder.
- So far, we have: 2 × (remaining factorization of 189)
2. Now, we move on to dividing the remaining factor, which is 189, by the smallest prime number again, 2. Once again, 189 divided by 2 is equal to 94.5, which is not a whole number. Therefore, 2 is not a factor of 189.
3. Next, we try dividing 189 by the next smallest prime number, which is 3. This time, 189 divided by 3 is equal to 63, with no remainder.
- So far, we have: 2 × 3 × (remaining factorization of 63)
4. Similarly, we divide the remaining factor, 63, by the same prime number, 3. 63 divided by 3 is equal to 21, with no remainder.
- So far, we have: 2 × 3 × 3 × (remaining factorization of 21)
5. Continuing the process, we divide the remaining factor, 21, by the smallest prime number, 2. But 21 is not divisible by 2 evenly.
6. We move on to dividing 21 by the next smallest prime number, 3. 21 divided by 3 is equal to 7, with no remainder.
- So far, we have: 2 × 3 × 3 × 7
7. At this point, we have found that the factors are all prime numbers. Thus, the final factorization of 378 is:
- 2 × 3 × 3 × 7
Therefore, the whole number 378 can be expressed as a product of prime numbers with exponents as 2 × 3^2 × 7.
To express the whole number 378 as a product of prime numbers with exponents, we need to factorize it into its prime factors. Here's how you can do it:
1. Start by dividing the number by the smallest prime number, which is 2. If 378 is divisible by 2, you can write it as "2 times the quotient."
378 ÷ 2 = 189
So, we can write 378 as 2 * 189.
2. Repeat the process with the quotient. Divide it by the smallest prime number, which is 2.
189 ÷ 2 = 94.5 (Note: We get a decimal, so 189 is not divisible by 2.)
3. Since 189 is not divisible by 2, we move on to the next prime number, which is 3. Check if 189 is divisible by 3.
189 ÷ 3 = 63
So, we can write 189 as 3 * 63.
4. Divide 63 by the smallest prime number, which is 3.
63 ÷ 3 = 21
So, we can write 63 as 3 * 21.
5. Now, divide 21 by the smallest prime number, which is 3.
21 ÷ 3 = 7
So, we can write 21 as 3 * 7.
6. We have reached a prime factor (7), and we can't divide it any further, so we stop here.
Putting it all together, we can express 378 as a product of prime numbers:
378 = 2 * 3 * 3 * 3 * 7
Therefore, the answer is: 2 times 3 cubed times 7, which is the same as 2 * 33 * 7.