Expressing Numbers As Products With Common Factors

Hey guys! Today, we're diving into a cool math problem: expressing numbers as products using common factors. This might sound a bit complex, but trust me, it's a super handy skill to have in your mathematical toolkit. We'll break down several examples step by step, so you can see exactly how it's done. We will focus on identifying the common factors, factoring them out, and simplifying the expressions to their final product form. Understanding these techniques will not only help you solve similar problems but also enhance your overall problem-solving abilities in mathematics. So, let's get started and make math a bit more fun and understandable!

Understanding Common Factors

Before we jump into solving the problems, let's make sure we're all on the same page about what a common factor is. In simple terms, a common factor is a number that divides exactly into two or more numbers. When we're dealing with expressions involving exponents, like the ones we have here, we look for the term with the lowest exponent as our common factor. This is because we can factor it out from all the other terms. Identifying and utilizing common factors is a fundamental technique in simplifying complex expressions and is a cornerstone of algebraic manipulation. Understanding this concept thoroughly can significantly enhance your ability to solve a wide range of mathematical problems, especially those involving exponents and polynomials. The process involves not just recognizing the commonality but also strategically using it to reduce the expression to a more manageable and understandable form, which is essential for further calculations or simplifications.

Think of it like this: if you have an expression like ax + ay, the common factor is a. We can factor it out to get a(x + y). We will apply this same principle to more complex expressions involving exponents. In the examples we are about to tackle, the common factors will often be exponential terms. For instance, when you encounter terms like 4³⁵, 4³⁶, and 4³⁴, the common factor will likely involve the lowest exponent, 4³⁴. This is because each term can be expressed as a multiple of 4³⁴, allowing us to factor it out and simplify the expression. Mastering this technique is particularly useful in various areas of mathematics, including algebra and calculus, where simplification is a crucial step in solving complex problems. By identifying the common factors, we can transform seemingly intricate expressions into more straightforward forms, making them easier to work with and leading to efficient solutions.

Problem c) 7 × 4³⁵ + 2 × 4³⁶ - 4³⁴

Let's tackle the first problem: 7 × 4³⁵ + 2 × 4³⁶ - 4³⁴. The first step is to identify the smallest exponent, which in this case is 4³⁴. We're going to rewrite the other terms so they include 4³⁴. This will help us factor it out easily. To rewrite the terms, we need to express 4³⁵ and 4³⁶ as products involving 4³⁴. Remember that a^(m+n) = a^m × a^n. So, 4³⁵ can be written as 4³⁴ × 4¹, and 4³⁶ can be written as 4³⁴ × 4². This transformation is crucial because it allows us to see the common factor of 4³⁴ in each term of the expression. By manipulating the exponents in this way, we prepare the expression for factorization, which is the next step in simplifying the problem. This process not only makes the expression easier to work with but also highlights the underlying structure, making it clear how the common factor can be extracted.

Now we can rewrite the original expression: 7 × (4³⁴ × 4¹) + 2 × (4³⁴ × 4²) - 4³⁴. Notice how 4³⁴ now appears in each term. This is exactly what we wanted! Next, we simplify the terms inside the parentheses. We know that 4¹ = 4 and 4² = 16, so the expression becomes: 7 × 4³⁴ × 4 + 2 × 4³⁴ × 16 - 4³⁴. Now we can multiply the constants: 7 × 4 = 28 and 2 × 16 = 32. So, the expression is now: 28 × 4³⁴ + 32 × 4³⁴ - 4³⁴. This simplification step is crucial because it consolidates the terms, making it even clearer how the common factor can be applied. By performing these basic arithmetic operations, we reduce the expression to a form where the factorization process is straightforward and intuitive, which is essential for solving the problem efficiently.

Now we factor out the common factor 4³⁴: 4³⁴(28 + 32 - 1). Notice that we write -1 because the last term was -4³⁴, which is the same as -1 × 4³⁴. This is a common mistake people make, so pay close attention to it! This step is the heart of the problem-solving process. By factoring out the common factor, we transform the expression from a sum of terms to a product, which is the goal of the problem. This not only simplifies the expression but also reveals its underlying structure, making it easier to understand and analyze. The careful attention to the -1 term is crucial because it maintains the mathematical integrity of the expression and ensures that the final result is accurate. Correctly factoring out the common factor is a fundamental skill in algebra and is essential for solving a wide range of mathematical problems.

Finally, we simplify the expression inside the parentheses: 28 + 32 - 1 = 59. So, the final answer is 59 × 4³⁴. And that's it! We've expressed the original number as a product using the common factor. This final simplification step consolidates all the previous steps into a single, concise answer. By performing the arithmetic operation within the parentheses, we arrive at the simplified product, which represents the original expression in its factored form. This not only completes the problem but also showcases the power of factorization in simplifying complex mathematical expressions. The final answer, 59 × 4³⁴, is a clear and understandable representation of the original expression, demonstrating the effectiveness of the common factor method in solving such problems.

Problem d) 8²⁰⁰ - 3 × 8¹⁹⁸ + 2 × 8¹⁹⁷

Let's move on to the next one: 8²⁰⁰ - 3 × 8¹⁹⁸ + 2 × 8¹⁹⁷. Just like before, we need to identify the smallest exponent, which is 8¹⁹⁷. We'll rewrite the other terms using this as our base. To do this, we express 8²⁰⁰ and 8¹⁹⁸ in terms of 8¹⁹⁷. Remember, we can use the rule a^(m+n) = a^m × a^n. So, 8²⁰⁰ can be written as 8¹⁹⁷ × 8³ and 8¹⁹⁸ can be written as 8¹⁹⁷ × 8¹. Rewriting the terms in this way is a crucial step in identifying the common factor and preparing the expression for simplification. By expressing each term as a product involving 8¹⁹⁷, we make it clear how the common factor can be factored out. This process not only simplifies the expression but also allows us to consolidate terms, making it easier to work with in subsequent steps. Understanding this manipulation of exponents is a fundamental skill in algebra and is essential for solving problems involving exponential expressions.

So, the expression becomes: (8¹⁹⁷ × 8³) - 3 × (8¹⁹⁷ × 8¹) + 2 × 8¹⁹⁷. We can now simplify the terms with the exponents. We know that 8³ = 512 and 8¹ = 8. The expression is now: 8¹⁹⁷ × 512 - 3 × 8¹⁹⁷ × 8 + 2 × 8¹⁹⁷. Next, we multiply the constants: 3 × 8 = 24. So, we have: 512 × 8¹⁹⁷ - 24 × 8¹⁹⁷ + 2 × 8¹⁹⁷. This step of simplifying the exponents and multiplying the constants is important as it prepares the expression for factoring. By performing these arithmetic operations, we reduce the complexity of the expression, making it easier to identify and factor out the common factor. This simplification allows us to consolidate terms and rewrite the expression in a more manageable form, which is crucial for the subsequent steps in solving the problem.

Now we factor out the common factor 8¹⁹⁷: 8¹⁹⁷(512 - 24 + 2). Notice how each term now clearly contributes to the expression inside the parentheses. Factoring out the common factor is a key step in simplifying the original expression. By isolating 8¹⁹⁷, we convert the expression from a series of terms to a product, which is the main goal of this type of problem. This not only simplifies the expression but also makes it easier to understand the relationship between the different components. The expression inside the parentheses now represents the sum and difference of constants, which can be easily computed to arrive at the final simplified form. This factorization step demonstrates a powerful technique in algebra, enabling us to transform complex expressions into more manageable forms.

Finally, we simplify the expression inside the parentheses: 512 - 24 + 2 = 490. So, the final answer is 490 × 8¹⁹⁷. Awesome! We've done it again. Simplifying the expression inside the parentheses completes the transformation of the original expression into its factored form. By performing this arithmetic operation, we consolidate the constants into a single number, which is then multiplied by the common factor we extracted earlier. The result, 490 × 8¹⁹⁷, is the simplified product, representing the original expression in a clear and concise manner. This final step not only provides the solution to the problem but also highlights the effectiveness of using common factors to simplify complex mathematical expressions.

Problem e) 46¹³⁵ + 5 × 46¹³⁶ - 46¹³⁴

Let's jump into problem e): 46¹³⁵ + 5 × 46¹³⁶ - 46¹³⁴. Can you guess the first step? Yep, we identify the smallest exponent, which is 46¹³⁴. We'll rewrite the other terms using this as our common factor. Rewriting the terms with a common factor is a fundamental strategy in simplifying expressions involving exponents. By identifying the smallest exponent, we can express other terms as products involving that base raised to the smallest exponent. This allows us to factor out the common factor, making the expression more manageable and easier to simplify. This initial step is crucial because it sets the stage for the subsequent steps of factorization and simplification, which ultimately lead to the solution. The ability to recognize and apply this strategy is essential for solving a wide range of mathematical problems, especially those encountered in algebra and calculus.

So, we rewrite 46¹³⁵ as 46¹³⁴ × 46¹ and 46¹³⁶ as 46¹³⁴ × 46². Now the expression looks like this: (46¹³⁴ × 46¹) + 5 × (46¹³⁴ × 46²) - 46¹³⁴. Let's simplify the terms inside the parentheses. We know 46¹ = 46 and 46² = 2116. So, we have: 46¹³⁴ × 46 + 5 × 46¹³⁴ × 2116 - 46¹³⁴. Now, we multiply the constants: 5 × 2116 = 10580. Our expression is now: 46 × 46¹³⁴ + 10580 × 46¹³⁴ - 46¹³⁴. Multiplying the constants and rewriting the terms is a crucial step in preparing the expression for factorization. This process simplifies the expression by consolidating numerical values, making it easier to identify and extract the common factor. By performing these arithmetic operations, we reduce the complexity of the expression, which streamlines the subsequent steps in solving the problem. This meticulous approach ensures accuracy and facilitates a clear understanding of the mathematical relationships within the expression.

Time to factor out the common factor 46¹³⁴: 46¹³⁴(46 + 10580 - 1). Don't forget that -1! Factoring out the common factor is a pivotal step in simplifying the original expression. By isolating 46¹³⁴, we convert the expression from a series of terms to a product, which is the primary goal of this type of problem. This not only simplifies the expression but also makes it easier to understand the relationship between the different components. Including the -1 ensures the mathematical integrity of the expression, as it accurately represents the subtraction of 46¹³⁴. This step is crucial for obtaining the correct final result and showcases the importance of attention to detail in mathematical problem-solving.

Now, let's simplify the expression inside the parentheses: 46 + 10580 - 1 = 10625. So, the final answer is 10625 × 46¹³⁴. We're on a roll! Simplifying the expression inside the parentheses is the final step in converting the original expression into its factored form. By performing this arithmetic operation, we consolidate the constants into a single number, which is then multiplied by the common factor we extracted earlier. The result, 10625 × 46¹³⁴, represents the simplified product and provides the solution to the problem. This step not only completes the mathematical process but also demonstrates the power and efficiency of using common factors to simplify complex expressions.

Problem f) 2¹⁰² - 2¹⁰¹ - 2¹⁰⁰ - 2⁹⁹

Last but not least, let's tackle problem f): 2¹⁰² - 2¹⁰¹ - 2¹⁰⁰ - 2⁹⁹. You know the drill by now! We find the smallest exponent, which is 2⁹⁹, and rewrite the other terms. Identifying the smallest exponent is the first crucial step in simplifying expressions with exponential terms. By recognizing 2⁹⁹ as the smallest exponent in the given expression, we set the foundation for rewriting other terms as multiples of 2⁹⁹. This approach allows us to factor out the common factor effectively, which is a fundamental strategy for simplifying complex algebraic expressions. This initial identification is essential for the subsequent steps of factorization and simplification, ultimately leading to a more manageable and understandable form of the expression.

We rewrite 2¹⁰² as 2⁹⁹ × 2³, 2¹⁰¹ as 2⁹⁹ × 2², and 2¹⁰⁰ as 2⁹⁹ × 2¹. So, the expression becomes: (2⁹⁹ × 2³) - (2⁹⁹ × 2²) - (2⁹⁹ × 2¹) - 2⁹⁹. Now, let's simplify the terms with the exponents. We know 2³ = 8, 2² = 4, and 2¹ = 2. So, we have: 8 × 2⁹⁹ - 4 × 2⁹⁹ - 2 × 2⁹⁹ - 2⁹⁹. Simplifying the exponents is a critical step in preparing the expression for factorization. By evaluating the powers of 2, we reduce the complexity of the terms, making it easier to identify and extract the common factor. This process not only simplifies the expression but also clarifies the mathematical relationships between the terms, which is essential for accurate and efficient problem-solving. The simplification of exponents allows us to consolidate terms and rewrite the expression in a more manageable form, setting the stage for the subsequent factorization.

Now we factor out the common factor 2⁹⁹: 2⁹⁹(8 - 4 - 2 - 1). Notice the -1 again! Factoring out the common factor is the central step in simplifying the given expression. By isolating 2⁹⁹, we transform the expression from a series of terms to a product, which is a key objective in simplifying mathematical expressions. The inclusion of -1 is crucial for maintaining the mathematical integrity of the expression, as it accurately accounts for the subtraction of 2⁹⁹. This step not only simplifies the expression but also highlights the underlying structure, making it easier to understand and analyze. The accurate factoring and inclusion of all terms, including the constant -1, are essential for arriving at the correct final result.

Finally, we simplify the expression inside the parentheses: 8 - 4 - 2 - 1 = 1. So, the final answer is 1 × 2⁹⁹, which is just 2⁹⁹. We nailed it! Simplifying the expression inside the parentheses completes the transformation of the original expression into its simplest factored form. By performing the arithmetic operations, we consolidate the constants into a single number, which is then multiplied by the common factor we extracted earlier. The result, 2⁹⁹, represents the simplified product and provides the final solution to the problem. This step not only concludes the mathematical process but also underscores the effectiveness of using common factors to reduce complex expressions to their simplest forms.

Conclusion

So, there you have it! We've successfully expressed several numbers as products using common factors. The key takeaway here is to always identify the smallest exponent and use that as your guide. Factoring out the common factor is a powerful technique that simplifies complex expressions and makes them easier to work with. Remember to practice these steps, and you'll become a pro at simplifying expressions in no time! Understanding the concept of common factors and mastering the technique of factoring them out is essential for simplifying mathematical expressions efficiently. This method not only makes problems more manageable but also provides a clear and structured approach to solving them. Consistent practice with these steps will undoubtedly improve your ability to tackle complex expressions and enhance your overall mathematical proficiency.

Keep practicing, guys, and you'll get the hang of it! Math can be fun, especially when you have the right tools and techniques at your disposal. Continue to explore different types of problems and apply these strategies to strengthen your understanding. The more you practice, the more confident and proficient you'll become in simplifying expressions and solving mathematical challenges. So, embrace the process, enjoy the journey, and keep honing your skills to unlock the full potential of your mathematical abilities!