Write Down The LCM Of The Following Numbers. Give Your Answer In Exponential Form.$\[ \begin{align*} A &= 2^9 \times 3^6 \times 5^4 \times 11^3 \\ B &= 2^2 \times 3^7 \times 7^2 \\ C &= 2^7 \times 3^3 \times 13^3 \end{align*} \\]

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Introduction

In mathematics, the least common multiple (LCM) is the smallest multiple that is exactly divisible by each of the numbers in a given set. When dealing with numbers in exponential form, finding the LCM can be a bit more complex. In this article, we will explore how to find the LCM of three numbers given in exponential form.

Understanding Exponential Form

Before we dive into finding the LCM, let's quickly review what exponential form means. Exponential form is a way of expressing numbers as a product of a base and an exponent. For example, the number 2^3 can be read as "2 to the power of 3" or "2 cubed." In this form, the base is 2 and the exponent is 3.

The Numbers in Exponential Form

We are given three numbers in exponential form:

  • A = 2^9 × 3^6 × 5^4 × 11^3
  • B = 2^2 × 3^7 × 7^2
  • C = 2^7 × 3^3 × 13^3

Step 1: Identify the Prime Factors

To find the LCM, we need to identify the prime factors of each number. Prime factors are the prime numbers that multiply together to give the original number.

  • A = 2^9 × 3^6 × 5^4 × 11^3
    • Prime factors: 2, 3, 5, 11
  • B = 2^2 × 3^7 × 7^2
    • Prime factors: 2, 3, 7
  • C = 2^7 × 3^3 × 13^3
    • Prime factors: 2, 3, 13

Step 2: Determine the Highest Power of Each Prime Factor

To find the LCM, we need to determine the highest power of each prime factor that appears in any of the numbers.

  • 2: The highest power of 2 is 9 (from A).
  • 3: The highest power of 3 is 7 (from B).
  • 5: The highest power of 5 is 4 (from A).
  • 7: The highest power of 7 is 2 (from B).
  • 11: The highest power of 11 is 3 (from A).
  • 13: The highest power of 13 is 3 (from C).

Step 3: Multiply the Prime Factors with the Highest Powers

Now that we have determined the highest power of each prime factor, we can multiply them together to find the LCM.

LCM = 2^9 × 3^7 × 5^4 × 7^2 × 11^3 × 13^3

Simplifying the LCM

We can simplify the LCM by combining the exponents of the same base.

LCM = 2^9 × 3^7 × 5^4 × 7^2 × 11^3 × 13^3

Conclusion

In this article, we have learned how to find the least common multiple (LCM) of numbers in exponential form. We identified the prime factors of each number, determined the highest power of each prime factor, and multiplied them together to find the LCM. The LCM of the given numbers is:

2^9 × 3^7 × 5^4 × 7^2 × 11^3 × 13^3

Final Answer

Q: What is the least common multiple (LCM) of numbers in exponential form?

A: The LCM of numbers in exponential form is the smallest multiple that is exactly divisible by each of the numbers in a given set.

Q: How do I find the LCM of numbers in exponential form?

A: To find the LCM of numbers in exponential form, you need to identify the prime factors of each number, determine the highest power of each prime factor, and multiply them together.

Q: What are prime factors?

A: Prime factors are the prime numbers that multiply together to give the original number.

Q: How do I identify the prime factors of a number in exponential form?

A: To identify the prime factors of a number in exponential form, you need to look at the base and the exponent. For example, the number 2^3 has a base of 2 and an exponent of 3.

Q: What is the highest power of a prime factor?

A: The highest power of a prime factor is the largest exponent that appears in any of the numbers.

Q: How do I determine the highest power of each prime factor?

A: To determine the highest power of each prime factor, you need to compare the exponents of each prime factor in each number.

Q: Can I simplify the LCM?

A: Yes, you can simplify the LCM by combining the exponents of the same base.

Q: What is an example of simplifying the LCM?

A: For example, if the LCM is 2^9 × 3^7 × 5^4 × 7^2 × 11^3 × 13^3, you can simplify it by combining the exponents of the same base:

2^9 × 3^7 × 5^4 × 7^2 × 11^3 × 13^3 = 2^9 × 3^7 × 5^4 × 7^2 × 11^3 × 13^3

Q: How do I know if the LCM is correct?

A: To know if the LCM is correct, you need to check if it is exactly divisible by each of the numbers in the given set.

Q: What are some common mistakes to avoid when finding the LCM?

A: Some common mistakes to avoid when finding the LCM include:

  • Not identifying the prime factors of each number
  • Not determining the highest power of each prime factor
  • Not multiplying the prime factors with the highest powers together
  • Not simplifying the LCM

Q: Can I use a calculator to find the LCM?

A: Yes, you can use a calculator to find the LCM. However, it is always a good idea to double-check your work to ensure that the LCM is correct.

Q: How do I apply the concept of LCM to real-life situations?

A: The concept of LCM can be applied to real-life situations such as:

  • Finding the least common multiple of two or more numbers in a recipe
  • Determining the smallest multiple that is exactly divisible by each of the numbers in a set of data
  • Finding the least common multiple of two or more numbers in a musical composition

Conclusion

In this article, we have answered some frequently asked questions about finding the least common multiple (LCM) of numbers in exponential form. We have covered topics such as identifying prime factors, determining the highest power of each prime factor, and simplifying the LCM. We hope that this article has been helpful in understanding the concept of LCM and how to apply it to real-life situations.