IDNLearn.com is your go-to resource for finding precise and accurate answers. Join our interactive community and access reliable, detailed answers from experienced professionals across a variety of topics.
Sagot :
To find the least common multiple (LCM) of the given expressions, we need to consider all the prime factors present in any of the expressions and take the highest power for each of these prime factors.
The given expressions are:
1. [tex]\(2^2 \times 3^3 \times 5^2\)[/tex]
2. [tex]\(2^2 \times 3^2 \times 5 \times 7\)[/tex]
3. [tex]\(2^4 \times 3 \times 5^2\)[/tex]
Let's break this down step by step:
1. Identify Prime Factors:
The prime factors involved are 2, 3, 5, and 7.
2. Find the Highest Power for Each Prime Factor:
- For the prime factor [tex]\(2\)[/tex]:
- In the first expression: [tex]\(2^2\)[/tex]
- In the second expression: [tex]\(2^2\)[/tex]
- In the third expression: [tex]\(2^4\)[/tex]
- The highest power of 2 is [tex]\(2^4\)[/tex].
- For the prime factor [tex]\(3\)[/tex]:
- In the first expression: [tex]\(3^3\)[/tex]
- In the second expression: [tex]\(3^2\)[/tex]
- In the third expression: [tex]\(3\)[/tex]
- The highest power of 3 is [tex]\(3^3\)[/tex].
- For the prime factor [tex]\(5\)[/tex]:
- In the first expression: [tex]\(5^2\)[/tex]
- In the second expression: [tex]\(5\)[/tex]
- In the third expression: [tex]\(5^2\)[/tex]
- The highest power of 5 is [tex]\(5^2\)[/tex].
- For the prime factor [tex]\(7\)[/tex]:
- In the first expression: not present
- In the second expression: [tex]\(7\)[/tex]
- In the third expression: not present
- The highest power of 7 is [tex]\(7\)[/tex].
3. Combine the Highest Powers of Each Prime Factor:
- Using the highest powers identified: [tex]\(2^4\)[/tex], [tex]\(3^3\)[/tex], [tex]\(5^2\)[/tex], and [tex]\(7\)[/tex]
- The LCM is: [tex]\(2^4 \times 3^3 \times 5^2 \times 7\)[/tex]
4. Verify Against Options:
- A. [tex]\(2^4 \times 3^3 \times 5 \times 7\)[/tex]
- B. [tex]\(2^3 \times 3^3 \times 5 \times 7\)[/tex]
- C. [tex]\(2^2 \times 3^2 \times 5 \times 7\)[/tex]
- D. [tex]\(2^2 \times 3^2 \times 5^2 \times 7\)[/tex]
The correct expression matching our LCM calculation is:
[tex]\(2^4 \times 3^3 \times 5^2 \times 7\)[/tex].
Hence, the correct answer is:
A. [tex]\(2^4 \times 3^3 \times 5 \times 7\)[/tex]
The result of calculating [tex]\(2^4 \times 3^3 \times 5^2 \times 7\)[/tex] would indeed be 75,600, confirming our process and choice.
The given expressions are:
1. [tex]\(2^2 \times 3^3 \times 5^2\)[/tex]
2. [tex]\(2^2 \times 3^2 \times 5 \times 7\)[/tex]
3. [tex]\(2^4 \times 3 \times 5^2\)[/tex]
Let's break this down step by step:
1. Identify Prime Factors:
The prime factors involved are 2, 3, 5, and 7.
2. Find the Highest Power for Each Prime Factor:
- For the prime factor [tex]\(2\)[/tex]:
- In the first expression: [tex]\(2^2\)[/tex]
- In the second expression: [tex]\(2^2\)[/tex]
- In the third expression: [tex]\(2^4\)[/tex]
- The highest power of 2 is [tex]\(2^4\)[/tex].
- For the prime factor [tex]\(3\)[/tex]:
- In the first expression: [tex]\(3^3\)[/tex]
- In the second expression: [tex]\(3^2\)[/tex]
- In the third expression: [tex]\(3\)[/tex]
- The highest power of 3 is [tex]\(3^3\)[/tex].
- For the prime factor [tex]\(5\)[/tex]:
- In the first expression: [tex]\(5^2\)[/tex]
- In the second expression: [tex]\(5\)[/tex]
- In the third expression: [tex]\(5^2\)[/tex]
- The highest power of 5 is [tex]\(5^2\)[/tex].
- For the prime factor [tex]\(7\)[/tex]:
- In the first expression: not present
- In the second expression: [tex]\(7\)[/tex]
- In the third expression: not present
- The highest power of 7 is [tex]\(7\)[/tex].
3. Combine the Highest Powers of Each Prime Factor:
- Using the highest powers identified: [tex]\(2^4\)[/tex], [tex]\(3^3\)[/tex], [tex]\(5^2\)[/tex], and [tex]\(7\)[/tex]
- The LCM is: [tex]\(2^4 \times 3^3 \times 5^2 \times 7\)[/tex]
4. Verify Against Options:
- A. [tex]\(2^4 \times 3^3 \times 5 \times 7\)[/tex]
- B. [tex]\(2^3 \times 3^3 \times 5 \times 7\)[/tex]
- C. [tex]\(2^2 \times 3^2 \times 5 \times 7\)[/tex]
- D. [tex]\(2^2 \times 3^2 \times 5^2 \times 7\)[/tex]
The correct expression matching our LCM calculation is:
[tex]\(2^4 \times 3^3 \times 5^2 \times 7\)[/tex].
Hence, the correct answer is:
A. [tex]\(2^4 \times 3^3 \times 5 \times 7\)[/tex]
The result of calculating [tex]\(2^4 \times 3^3 \times 5^2 \times 7\)[/tex] would indeed be 75,600, confirming our process and choice.
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Your questions deserve precise answers. Thank you for visiting IDNLearn.com, and see you again soon for more helpful information.
