how can the whole number 378 be expressed as a product of prime numbers with exponents
The prime factorization of the whole number 378 can be expressed as:
378 = 2^1 × 3^3 × 7^1
To express the whole number 378 as a product of prime numbers with exponents, we need to find the prime factorization of 378.
Step 1: Find the smallest prime number that divides 378 evenly. In this case, it is 2. Divide 378 by 2 to get 189.
Step 2: Repeat the process with the new number obtained, which is 189. The smallest prime number that divides it evenly is 3. Divide 189 by 3 to get 63.
Step 3: Repeat the process with the new number obtained, which is 63. The smallest prime number that divides it evenly is 3. Divide 63 by 3 to get 21.
Step 4: Repeat the process with the new number obtained, which is 21. The smallest prime number that divides it evenly is 3. Divide 21 by 3 to get 7.
Step 5: Now we have reached a prime number, 7. It cannot be divided any further.
The prime factorization of 378 is therefore 2^1 * 3^3 * 7^1.
To express the whole number 378 as a product of prime numbers with exponents, you can follow these steps:
Step 1: Find the prime factors of 378.
Start by dividing 378 by the smallest prime number, which is 2. If 378 is divisible by 2, divide it by 2 and continue dividing the quotient by 2 until you get a quotient that is not divisible by 2. Repeat this process for the other prime numbers: 3, 5, 7, 11, and so on. Keep dividing until the quotient is 1.
Step 2: Write the prime factors with exponents.
Write down the prime factors you found in step 1. If a prime factor occurs multiple times, you can use exponents to indicate the number of times it appears. For example, if 2 appears three times, you would write it as 2^3. Repeat this for all prime factors.
Let's calculate the prime factorization of 378:
Dividing 378 by 2, we get: 378 ÷ 2 = 189.
Dividing 189 by 3, we get: 189 ÷ 3 = 63.
Dividing 63 by 3, we get: 63 ÷ 3 = 21.
Dividing 21 by 7, we get: 21 ÷ 7 = 3.
Now, we cannot divide 3 any further since it is already a prime number.
So, the prime factorization of 378 is: 2^1 × 3^3 × 7^1.
Therefore, the whole number 378 can be expressed as the product of prime numbers with exponents as: 2^1 × 3^3 × 7^1.