Answer :
Sure, let's go through the process of drawing a prime factor tree for 220 and writing its prime decomposition in index form.
### Step-by-Step Solution:
1. Start with the number 220:
- 220
2. Find two factors of 220:
- Choose one factor to be a prime number if possible. A good start would be to check for divisibility by small prime numbers like 2, 3, 5, etc.
- 220 is an even number, so it's divisible by 2.
[tex]\[
220 = 2 \times 110
\][/tex]
- Write it down:
- 220
- / \
- 2 110
3. Factor 110:
- Again, 110 is even and divisible by 2.
[tex]\[
110 = 2 \times 55
\][/tex]
- Update the tree:
- 220
- / \
- 2 110
- / \
- 2 55
4. Factor 55:
- 55 is not even, so it's not divisible by 2. Check the next prime, which is 3, but 55 is not divisible by 3 either.
- Check the next prime, 5. 55 is divisible by 5.
[tex]\[
55 = 5 \times 11
\][/tex]
- Update the tree:
- 220
- / \
- 2 110
- / \
- 2 55
- / \
- 5 11
5. Factor 11:
- 11 is a prime number and cannot be factored further.
[tex]\[
11 = 11 \times 1
\][/tex]
- Complete the tree:
- 220
- / \
- 2 110
- / \
- 2 55
- / \
- 5 11
6. Collect all the prime factors:
- From the tree, the prime factors of 220 are: 2, 2, 5, and 11.
7. Write the prime decomposition in index form:
- The prime factorization of 220 involves:
- Two 2’s (which means [tex]\(2^2\)[/tex])
- One 5 (which means [tex]\(5^1\)[/tex])
- One 11 (which means [tex]\(11^1\)[/tex])
[tex]\[
220 = 2^2 \times 5^1 \times 11^1
\][/tex]
### Final Answer:
The prime decomposition of 220 in index form is [tex]\(220 = 2^2 \times 5^1 \times 11^1\)[/tex].
### Step-by-Step Solution:
1. Start with the number 220:
- 220
2. Find two factors of 220:
- Choose one factor to be a prime number if possible. A good start would be to check for divisibility by small prime numbers like 2, 3, 5, etc.
- 220 is an even number, so it's divisible by 2.
[tex]\[
220 = 2 \times 110
\][/tex]
- Write it down:
- 220
- / \
- 2 110
3. Factor 110:
- Again, 110 is even and divisible by 2.
[tex]\[
110 = 2 \times 55
\][/tex]
- Update the tree:
- 220
- / \
- 2 110
- / \
- 2 55
4. Factor 55:
- 55 is not even, so it's not divisible by 2. Check the next prime, which is 3, but 55 is not divisible by 3 either.
- Check the next prime, 5. 55 is divisible by 5.
[tex]\[
55 = 5 \times 11
\][/tex]
- Update the tree:
- 220
- / \
- 2 110
- / \
- 2 55
- / \
- 5 11
5. Factor 11:
- 11 is a prime number and cannot be factored further.
[tex]\[
11 = 11 \times 1
\][/tex]
- Complete the tree:
- 220
- / \
- 2 110
- / \
- 2 55
- / \
- 5 11
6. Collect all the prime factors:
- From the tree, the prime factors of 220 are: 2, 2, 5, and 11.
7. Write the prime decomposition in index form:
- The prime factorization of 220 involves:
- Two 2’s (which means [tex]\(2^2\)[/tex])
- One 5 (which means [tex]\(5^1\)[/tex])
- One 11 (which means [tex]\(11^1\)[/tex])
[tex]\[
220 = 2^2 \times 5^1 \times 11^1
\][/tex]
### Final Answer:
The prime decomposition of 220 in index form is [tex]\(220 = 2^2 \times 5^1 \times 11^1\)[/tex].