3. Write each number as a product of its prime factors (prime factorization).
Express your answers in exponential form.
(a)98 (b) 200 (c) 140 (d) 80
a) 98 = 2^1 * 7^2
b) 200 = 2^3 * 5^2
c) 140 = 2^2 * 5^1 * 7^1
d) 80 = 2^4 * 5^0 * 7^0
(a) To write 98 as a product of its prime factors, we can start by dividing it by the smallest prime number, which is 2.
Step 1: Divide 98 by 2. 98 ÷ 2 = 49
Now we have 49. Since 49 is not divisible by 2, we move on to the next prime number, which is 3.
Step 2: Divide 49 by 3. 49 ÷ 3 = 16 remainder 1
Now we have 16. We keep dividing by prime numbers until we can no longer divide evenly.
Step 3: Divide 16 by 2. 16 ÷ 2 = 8
Step 4: Divide 8 by 2. 8 ÷ 2 = 4
Step 5: Divide 4 by 2. 4 ÷ 2 = 2
Finally, we have 2, which is a prime number and cannot be divided further.
As a result, the prime factorization of 98 is 2^1 × 7^2. (Exponential form)
(b) To write 200 as a product of its prime factors, we can again start by dividing it by 2.
Step 1: Divide 200 by 2. 200 ÷ 2 = 100
Now we have 100. We continue dividing by 2 until we can no longer divide evenly.
Step 2: Divide 100 by 2. 100 ÷ 2 = 50
Step 3: Divide 50 by 2. 50 ÷ 2 = 25
Now we have 25. Since 25 is not divisible by 2, we move on to the next prime number, which is 5.
Step 4: Divide 25 by 5. 25 ÷ 5 = 5
Finally, we have 5, which is a prime number and cannot be divided further.
As a result, the prime factorization of 200 is 2^3 × 5^2. (Exponential form)
(c) To write 140 as a product of its prime factors, we can again start by dividing it by 2.
Step 1: Divide 140 by 2. 140 ÷ 2 = 70
Now we have 70. We continue dividing by 2 until we can no longer divide evenly.
Step 2: Divide 70 by 2. 70 ÷ 2 = 35
Now we have 35. Since 35 is not divisible by 2, we move on to the next prime number, which is 5.
Step 3: Divide 35 by 5. 35 ÷ 5 = 7
Finally, we have 7, which is a prime number and cannot be divided further.
As a result, the prime factorization of 140 is 2^2 × 5^1 × 7^1. (Exponential form)
(d) To write 80 as a product of its prime factors, we can start by dividing it by 2.
Step 1: Divide 80 by 2. 80 ÷ 2 = 40
Now we have 40. We continue dividing by 2 until we can no longer divide evenly.
Step 2: Divide 40 by 2. 40 ÷ 2 = 20
Step 3: Divide 20 by 2. 20 ÷ 2 = 10
Now we have 10. Since 10 is not divisible by 2, we move on to the next prime number, which is 5.
Step 4: Divide 10 by 5. 10 ÷ 5 = 2
Finally, we have 2, which is a prime number and cannot be divided further.
As a result, the prime factorization of 80 is 2^4 × 5^1. (Exponential form)
To factorize a number into its prime factors, we need to find the prime numbers that divide the given number evenly.
(a) To find the prime factorization of 98:
Step 1: Divide the number by the smallest prime number, which is 2.
98 ÷ 2 = 49
Step 2: Now, try to divide the number again by 2:
49 is not divisible by 2.
Step 3: Move on to the next prime number, which is 3.
49 ÷ 3 = 16 remainder 1
Step 4: Since 16 is divisible by 2, divide it by 2:
16 ÷ 2 = 8
Step 5: 8 is divisible by 2, so divide it by 2 again:
8 ÷ 2 = 4
Step 6: 4 is divisible by 2, so divide it by 2 again:
4 ÷ 2 = 2
Therefore, the prime factorization of 98 is 2 × 7 × 7, or in exponential form: 2² × 7².
(b) To find the prime factorization of 200:
Step 1: Divide the number by the smallest prime number, which is 2.
200 ÷ 2 = 100
Step 2: Divide the result, 100, by 2 again:
100 ÷ 2 = 50
Step 3: Divide 50, by 2 again:
50 ÷ 2 = 25
Step 4: 25 is not divisible by 2, so try the next prime number, which is 3.
25 ÷ 5 = 5
Therefore, the prime factorization of 200 is 2³ × 5².
(c) To find the prime factorization of 140:
Step 1: Divide the number by the smallest prime number, which is 2.
140 ÷ 2 = 70
Step 2: Divide the result, 70, by 2 again:
70 ÷ 2 = 35
Step 3: Since 35 is not divisible by 2, try dividing by the next prime number, which is 3.
35 ÷ 5 = 7
Therefore, the prime factorization of 140 is 2² × 5 × 7.
(d) To find the prime factorization of 80:
Step 1: Divide the number by the smallest prime number, which is 2.
80 ÷ 2 = 40
Step 2: Divide the result, 40, by 2 again:
40 ÷ 2 = 20
Step 3: Divide the result, 20, by 2 again:
20 ÷ 2 = 10
Step 4: Divide the result, 10, by 2 again:
10 ÷ 2 = 5
Therefore, the prime factorization of 80 is 2⁴ × 5.