Calculating The Product Of 144 Multiplied By 2^-3, 12^-1, And 3^-2

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In the realm of mathematics, exploring the interplay between numbers and exponents often reveals fascinating results. This article delves into a specific problem involving the number 144 and its successive multiplication by 2^-3, 12^-1, and 3^-2. We will meticulously dissect the problem, applying the fundamental principles of exponents and fractions to arrive at the final product. Our journey will not only provide the solution but also illuminate the underlying mathematical concepts, fostering a deeper appreciation for the elegance of numerical operations.

Breaking Down the Problem

At the heart of our exploration lies the expression 144 * 2^-3 * 12^-1 * 3^-2. To unravel this, we must first understand the implications of negative exponents. A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent. For instance, 2^-3 is equivalent to 1 / (2^3), which equals 1/8. Similarly, 12^-1 translates to 1/12, and 3^-2 becomes 1 / (3^2), or 1/9. By grasping this fundamental concept, we pave the way for simplifying the expression and determining the final product.

Now that we have deciphered the meaning of the negative exponents, we can rewrite the expression as 144 * (1/8) * (1/12) * (1/9). This transformation allows us to visualize the problem as a series of multiplications involving fractions. To solve this, we can multiply the numerators (the top numbers) and the denominators (the bottom numbers) separately. The product of the numerators is simply 144 * 1 * 1 * 1 = 144, while the product of the denominators is 8 * 12 * 9 = 864. Thus, the expression simplifies to 144/864. However, this fraction is not in its simplest form, and we can further reduce it by finding the greatest common divisor (GCD) of 144 and 864.

To find the GCD, we can employ the prime factorization method. Prime factorization involves breaking down a number into its prime factors, which are prime numbers that divide the number evenly. The prime factorization of 144 is 2^4 * 3^2, while the prime factorization of 864 is 2^5 * 3^3. The GCD is found by taking the lowest power of each common prime factor. In this case, the common prime factors are 2 and 3, and their lowest powers are 2^4 and 3^2, respectively. Therefore, the GCD of 144 and 864 is 2^4 * 3^2 = 16 * 9 = 144. Dividing both the numerator and the denominator of the fraction 144/864 by the GCD, 144, we obtain the simplified fraction 1/6. This is the final product of the given expression.

Step-by-Step Solution

To solidify our understanding, let's walk through the solution step-by-step:

  1. Understand Negative Exponents: Recall that a^-n = 1 / a^n.
  2. Rewrite the Expression: Substitute the negative exponents with their reciprocal equivalents: 144 * (1/8) * (1/12) * (1/9).
  3. Multiply the Fractions: Multiply the numerators and denominators separately: (144 * 1 * 1 * 1) / (8 * 12 * 9) = 144/864.
  4. Find the Greatest Common Divisor (GCD): Determine the GCD of 144 and 864 using prime factorization or other methods. The GCD is 144.
  5. Simplify the Fraction: Divide both the numerator and denominator by the GCD: 144/864 = (144/144) / (864/144) = 1/6.

Therefore, the final product of 144 multiplied by 2^-3, 12^-1, and 3^-2 consecutively is 1/6.

Alternative Approach: Prime Factorization from the Start

An alternative approach to solving this problem involves utilizing prime factorization from the very beginning. This method offers a different perspective and can sometimes streamline the calculation process. Let's explore how it works.

  1. Prime Factorize Each Number: Break down each number in the expression into its prime factors:

    • 144 = 2^4 * 3^2
    • 2^-3 = 1 / (2^3)
    • 12^-1 = 1 / 12 = 1 / (2^2 * 3)
    • 3^-2 = 1 / (3^2)

  2. Rewrite the Expression with Prime Factors: Substitute the numbers with their prime factorizations: (2^4 * 3^2) * (1 / (2^3)) * (1 / (2^2 * 3)) * (1 / (3^2))

  3. Combine the Terms: Combine the terms with the same base by adding or subtracting the exponents. Remember that dividing by a number with an exponent is the same as multiplying by the number with the negative of that exponent: 2^(4 - 3 - 2) * 3^(2 - 1 - 2) = 2^(-1) * 3^(-1)

  4. Simplify: Rewrite the expression with positive exponents: (1 / 2^1) * (1 / 3^1) = (1/2) * (1/3)

  5. Multiply: Multiply the fractions: (1 * 1) / (2 * 3) = 1/6

This alternative approach, using prime factorization from the start, yields the same final product of 1/6. It demonstrates the flexibility and interconnectedness of mathematical concepts, allowing us to tackle problems from different angles.

Key Concepts and Principles

This problem highlights several key concepts and principles in mathematics:

  • Negative Exponents: A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent (a^-n = 1 / a^n). Understanding this principle is crucial for manipulating expressions involving negative exponents.
  • Prime Factorization: Prime factorization is the process of breaking down a number into its prime factors. It is a fundamental tool in number theory and has applications in simplifying fractions, finding the greatest common divisor (GCD), and solving various mathematical problems.
  • Greatest Common Divisor (GCD): The GCD of two or more numbers is the largest positive integer that divides all the numbers without leaving a remainder. Finding the GCD is essential for simplifying fractions and solving certain types of equations.
  • Fraction Multiplication: Multiplying fractions involves multiplying the numerators and the denominators separately. The resulting fraction can then be simplified by dividing both the numerator and denominator by their GCD.
  • Exponent Rules: The rules of exponents govern how exponents behave in various mathematical operations. These rules include the product rule (a^m * a^n = a^(m+n)), the quotient rule (a^m / a^n = a^(m-n)), and the power rule ((am)n = a^(m*n)).

By mastering these concepts and principles, you can confidently tackle a wide range of mathematical problems involving exponents, fractions, and prime factorization.

Real-World Applications

While the problem we solved may seem abstract, the concepts and skills involved have numerous real-world applications. Understanding exponents and fractions is fundamental in various fields, including:

  • Finance: Calculating compound interest, investment returns, and loan payments involves exponents and fractions.
  • Science: Scientific notation, which uses exponents to express very large or very small numbers, is essential in physics, chemistry, and astronomy. Understanding fractions is crucial for measurements, ratios, and proportions.
  • Engineering: Engineers use exponents and fractions in calculations related to structural design, electrical circuits, and fluid dynamics.
  • Computer Science: Binary numbers, which are the foundation of computer systems, are based on powers of 2. Understanding exponents and fractions is also important for data analysis and algorithm design.
  • Everyday Life: We encounter fractions and exponents in everyday situations such as cooking, measuring, and calculating discounts and percentages.

By developing a strong foundation in these mathematical concepts, you equip yourself with valuable tools for success in various academic and professional pursuits.

Conclusion

In this article, we have thoroughly examined the problem of multiplying 144 by 2^-3, 12^-1, and 3^-2 consecutively. We have not only arrived at the final product of 1/6 but also delved into the underlying mathematical concepts and principles that make the solution possible. We have explored the significance of negative exponents, the power of prime factorization, and the importance of simplifying fractions. Furthermore, we have highlighted the real-world applications of these concepts, demonstrating their relevance beyond the realm of abstract mathematics.

By mastering these fundamental mathematical skills, you empower yourself to tackle a wide array of problems and challenges in various fields. The journey through this problem serves as a testament to the elegance and interconnectedness of mathematics, inspiring a deeper appreciation for its power and versatility. As you continue your mathematical explorations, remember that every problem is an opportunity to learn, grow, and expand your understanding of the world around you.

  • What is a negative exponent? A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent. For example, a^-n = 1 / a^n.
  • How do you find the greatest common divisor (GCD) of two numbers? The GCD can be found using various methods, including prime factorization, the Euclidean algorithm, and listing factors. Prime factorization involves breaking down each number into its prime factors, and the GCD is found by taking the lowest power of each common prime factor.
  • Why is it important to simplify fractions? Simplifying fractions makes them easier to understand and compare. A simplified fraction is in its lowest terms, meaning that the numerator and denominator have no common factors other than 1.
  • What are the rules of exponents? The rules of exponents govern how exponents behave in various mathematical operations. Some key rules include the product rule (a^m * a^n = a^(m+n)), the quotient rule (a^m / a^n = a^(m-n)), and the power rule ((am)n = a^(m*n)).
  • Where can I learn more about exponents and fractions? You can learn more about exponents and fractions from various resources, including textbooks, online tutorials, educational websites, and math teachers or tutors. Practice solving problems regularly to solidify your understanding.