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What is the LCM of these index numbers?

[tex]2^2 \times 3^3 \times 5^2, \ 2^2 \times 3^2 \times 5 \times 7, \ \text{and} \ 2^4 \times 3 \times 5^2[/tex]

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]

Sagot :

GinnyAnswer
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.
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