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

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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 numbers are expressed in exponential form, finding the LCM becomes a straightforward process. 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 understand 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." This means that 2^3 is equal to 2 multiplied by itself 3 times, which equals 8.

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

Finding the LCM

To find the LCM of these numbers, we need to follow these steps:

  1. List the prime factors: Identify the prime factors of each number and their corresponding exponents.
  2. Identify the highest exponent: For each prime factor, identify the highest exponent among the three numbers.
  3. Multiply the highest exponents: Multiply the highest exponents of each prime factor to find the LCM.

Step 1: List the Prime Factors

Let's list the prime factors of each number:

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

Step 2: Identify the Highest Exponent

Now, let's identify the highest exponent of each prime factor:

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

Step 3: Multiply the Highest Exponents

Now, let's multiply the highest exponents of each prime factor:

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

Simplifying the LCM

To simplify the LCM, we can rewrite it in exponential form:

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

Conclusion

In this article, we learned how to find the least common multiple (LCM) of numbers given in exponential form. We followed a step-by-step process to identify the prime factors, highest exponents, and multiply them to find the LCM. The LCM of the given numbers is 2^9 × 3^7 × 5^4 × 7^2 × 11^3 × 13^3.

Example Use Case

Finding the LCM of numbers in exponential form has many practical applications in mathematics and real-world problems. For example, in music, the LCM of two or more time signatures can help determine the common time signature. In physics, the LCM of two or more frequencies can help determine the common frequency.

Common Mistakes to Avoid

When finding the LCM of numbers in exponential form, there are a few common mistakes to avoid:

  • Not listing the prime factors: Make sure to list the prime factors of each number and their corresponding exponents.
  • Not identifying the highest exponent: Make sure to identify the highest exponent of each prime factor.
  • Not multiplying the highest exponents: Make sure to multiply the highest exponents of each prime factor to find the LCM.

Final Thoughts

Frequently Asked Questions

In this article, we will answer some frequently asked questions about finding the least common multiple (LCM) of numbers in exponential form.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) 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, follow these steps:

  1. List the prime factors: Identify the prime factors of each number and their corresponding exponents.
  2. Identify the highest exponent: For each prime factor, identify the highest exponent among the numbers.
  3. Multiply the highest exponents: Multiply the highest exponents of each prime factor to find the LCM.

Q: What if I have a number with a negative exponent?

A: If you have a number with a negative exponent, you can rewrite it as a fraction with a positive exponent. For example, 2^(-3) can be rewritten as 1/2^3.

Q: What if I have a number with a zero exponent?

A: If you have a number with a zero exponent, it means that the number is equal to 1. For example, 2^0 = 1.

Q: Can I use the LCM to find the greatest common divisor (GCD)?

A: Yes, you can use the LCM to find the greatest common divisor (GCD). The GCD of two numbers is equal to the product of the two numbers divided by their LCM.

Q: What is the relationship between the LCM and the GCD?

A: The LCM and the GCD are related by the following equation:

LCM(a, b) × GCD(a, b) = a × b

Q: Can I use the LCM to solve real-world problems?

A: Yes, the LCM has many practical applications in mathematics and real-world problems. For example, in music, the LCM of two or more time signatures can help determine the common time signature. In physics, the LCM of two or more frequencies can help determine the common frequency.

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

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

  • Not listing the prime factors: Make sure to list the prime factors of each number and their corresponding exponents.
  • Not identifying the highest exponent: Make sure to identify the highest exponent of each prime factor.
  • Not multiplying the highest exponents: Make sure to multiply the highest exponents of each prime factor to find the LCM.

Q: How can I practice finding the LCM?

A: You can practice finding the LCM by working through examples and exercises. You can also use online resources and calculators to help you find the LCM.

Conclusion

Finding the least common multiple (LCM) of numbers in exponential form is a straightforward process that requires attention to detail and a clear understanding of prime factors and exponents. By following the steps outlined in this article and practicing with examples and exercises, you can become proficient in finding the LCM and apply it to real-world problems.

Example Problems

Here are some example problems to help you practice finding the LCM:

  • Find the LCM of 2^3 × 3^2 and 2^2 × 3^4.
  • Find the LCM of 5^3 × 7^2 and 5^2 × 7^4.
  • Find the LCM of 2^5 × 3^3 and 2^3 × 3^5.

Answer Key

Here are the answers to the example problems:

  • LCM = 2^3 × 3^4
  • LCM = 5^3 × 7^4
  • LCM = 2^5 × 3^5