how can the whole number 378 be expressed as a product of prime numbers with exponents
To express 378 as a product of prime numbers with exponents, we need to find its prime factorization.
Start by dividing 378 by the smallest prime number, which is 2.
378 ÷ 2 = 189
Now, divide 189 by 2 again.
189 ÷ 2 = 94.5
Since 94.5 is not a whole number, we move on to the next prime number, which is 3.
189 ÷ 3 = 63
Continuing with the next prime number, divide 63 by 3.
63 ÷ 3 = 21
The next prime number is 5, so divide 21 by 5.
21 ÷ 5 = 4.2
Since 4.2 is not a whole number, we move on to the next prime number, which is 7.
21 ÷ 7 = 3
Now, we cannot divide further because 3 is a prime number itself.
By multiplying all the divisors 2, 2, 3, 3, 7, and 3 together, we get the prime factorization of 378:
2 × 2 × 3 × 3 × 7 × 3 = 2^2 × 3^2 × 7 × 3 = 2^2 × 3^3 × 7
Therefore, 378 can be expressed as a product of prime numbers with exponents as 2^2 × 3^3 × 7.
To express the whole number 378 as a product of prime numbers with exponents, we need to factorize 378 into its prime factors. Here's how you can do it step-by-step:
1. Begin by dividing 378 by the smallest prime number, which is 2:
378 ÷ 2 = 189
2. Continue dividing the quotient, 189, by 2:
189 ÷ 2 = 94.5
3. Since 94.5 is not a whole number, move on to the next prime number, 3:
189 ÷ 3 = 63
4. Divide the quotient, 63, by 3:
63 ÷ 3 = 21
5. Again, divide the quotient, 21, by 3:
21 ÷ 3 = 7
6. Finally, divide the quotient, 7, by 7:
7 ÷ 7 = 1
At this point, we have reached the prime factorization of 378. Now, let's express it as a product of prime numbers with exponents:
378 = 2^1 * 3^3 * 7^1
Therefore, the whole number 378 can be expressed as a product of prime numbers with exponents as 2 raised to the power of 1, multiplied by 3 raised to the power of 3, multiplied by 7 raised to the power of 1.
To express the whole number 378 as a product of prime numbers with exponents, we need to find the prime factorization of 378.
The prime factorization is the representation of a number as a product of its prime factors. To find the prime factorization of a number, we divide the number by the smallest prime number possible, and repeat the process with the quotient until we reach a quotient of 1.
Let's start by dividing 378 by 2:
378 ÷ 2 = 189
189 is not divisible by 2, so we try the next prime number, which is 3:
189 ÷ 3 = 63
63 is also divisible by 3:
63 ÷ 3 = 21
21 is not divisible by 3, so we move on to the next prime number, 5:
21 ÷ 5 = 4 remainder 1
Next, we try dividing by 7:
21 ÷ 7 = 3
Now, we can see that 3 is prime, so the process ends here.
To summarize, the prime factorization of 378 is:
2 × 3 × 3 × 3 × 7
To express this using exponents, we can write:
2^1 × 3^3 × 7^1
So, the whole number 378 can be expressed as a product of prime numbers with exponents as 2^1 × 3^3 × 7^1.