Sure, let's work through each part of the question step-by-step.
### Part (a)
We need to write 600 as the product of its prime factors in index form.
1. Prime Factorization of 600:
- 600 can be factored into prime numbers in the following steps:
- 600 ÷ 2 = 300
- 300 ÷ 2 = 150
- 150 ÷ 2 = 75
- 75 ÷ 3 = 25
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1
So, the prime factors of 600 are 2, 3, and 5.
2. Count the Occurrences:
- The number 2 appears 3 times.
- The number 3 appears 1 time.
- The number 5 appears 2 times.
3. Express in Index Form:
- Therefore, 600 can be expressed as:
[tex]\( 600 = 2^3 \times 3^1 \times 5^2 \)[/tex]
### Part (b)
We need to determine the Highest Common Factor (HCF) of 600 and 1050.
1. Prime Factorization of 1050:
- 1050 can be factored into prime numbers in the following steps:
- 1050 ÷ 2 = 525
- 525 ÷ 3 = 175
- 175 ÷ 5 = 35
- 35 ÷ 5 = 7
- 7 ÷ 7 = 1
So, the prime factors of 1050 are 2, 3, 5, and 7.
- Factors: [tex]\( 1050 = 2^1 \times 3^1 \times 5^2 \times 7^1 \)[/tex]
2. Prime Factorization of 600:
- Recall from part (a): [tex]\( 600 = 2^3 \times 3^1 \times 5^2 \)[/tex]
3. Common Factors with the Lowest Powers:
- Common prime factors of 600 and 1050 are 2, 3, and 5.
- Minimum power of 2 in both numbers: [tex]\( \text{min}(2^3, 2^1) = 2^1 \)[/tex]
- Minimum power of 3 in both numbers: [tex]\( \text{min}(3^1, 3^1) = 3^1 \)[/tex]
- Minimum power of 5 in both numbers: [tex]\( \text{min}(5^2, 5^2) = 5^2 \)[/tex]
4. HCF Calculation:
- Multiply these lowest powers together to find the HCF:
[tex]\( \text{HCF} = 2^1 \times 3^1 \times 5^2 = 2 \times 3 \times 25 = 150 \)[/tex]
Answers:
1. Prime factorization of 600 in index form:
[tex]\( 600 = 2^3 \times 3^1 \times 5^2 \)[/tex]
2. Highest Common Factor (HCF) of 600 and 1050:
[tex]\( \text{HCF} = 150 \)[/tex]
