Evaluate (109)² Using A Suitable Identity. How To Solve This Mathematical Problem?

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Evaluating squares of numbers, especially larger ones, can seem daunting at first. However, by leveraging suitable algebraic identities, we can simplify these calculations significantly. In this comprehensive guide, we will delve into the process of evaluating (109)² using the power of algebraic identities. This method not only makes the calculation easier but also provides a deeper understanding of mathematical principles. Let's explore how we can break down this seemingly complex problem into manageable steps and arrive at the solution with clarity and precision. This approach is particularly useful in situations where calculators are not readily available, or when the focus is on demonstrating a strong grasp of fundamental mathematical concepts. Understanding and applying these identities can greatly enhance your problem-solving skills and mathematical fluency.

Understanding Algebraic Identities

Before we dive into evaluating (109)², it's crucial to understand what algebraic identities are and why they are so useful. Algebraic identities are equations that hold true for all values of the variables involved. They provide us with ready-made formulas to simplify expressions and solve problems more efficiently. These identities are like shortcuts in the world of mathematics, allowing us to bypass lengthy calculations and arrive at the answer quickly. One of the most commonly used identities in such scenarios is the (a + b)² identity, which states that (a + b)² = a² + 2ab + b². This identity is particularly useful when dealing with numbers that can be easily broken down into the sum of two simpler numbers. Other useful identities include (a - b)² = a² - 2ab + b² and (a + b)(a - b) = a² - b², each serving specific purposes in simplifying mathematical expressions. The ability to recognize and apply these identities is a fundamental skill in algebra, enhancing both speed and accuracy in problem-solving. In the context of evaluating squares, these identities provide a structured approach to breaking down complex numbers into simpler components, making the calculations more manageable and less prone to error. Mastering these identities not only aids in numerical computations but also strengthens the conceptual understanding of algebraic relationships.

Choosing the Right Identity for (109)²

When it comes to evaluating (109)², the key lies in choosing the right algebraic identity. Given the structure of the number 109, the most suitable identity to apply is the (a + b)² identity, which, as we discussed, is (a + b)² = a² + 2ab + b². This identity is perfect for this scenario because 109 can be easily expressed as the sum of two numbers: 100 and 9. By breaking down 109 into 100 + 9, we can directly apply the (a + b)² identity, where 'a' would be 100 and 'b' would be 9. This approach transforms the problem from a straightforward squaring operation into a series of simpler calculations involving squaring and multiplication, which are often easier to manage mentally or on paper. The elegance of using this identity lies in its ability to leverage the simplicity of squaring multiples of 10 (like 100) and smaller numbers, thereby reducing the complexity of the overall calculation. Choosing the correct identity is a critical step in simplifying mathematical problems, and in this case, the (a + b)² identity provides an efficient and accurate pathway to the solution. This method underscores the importance of recognizing patterns and structures within mathematical problems, allowing for the strategic application of algebraic tools.

Applying the (a + b)² Identity to Evaluate (109)²

Now, let's apply the (a + b)² identity to evaluate (109)². As we've established, we can express 109 as 100 + 9. Therefore, in our identity (a + b)² = a² + 2ab + b², 'a' is 100 and 'b' is 9. Substituting these values into the identity, we get (100 + 9)² = 100² + 2(100)(9) + 9². This breaks down the original problem into three simpler calculations: squaring 100, multiplying 2 by 100 and 9, and squaring 9. The first part, 100², is straightforward: it equals 10,000. The second part, 2(100)(9), is also manageable: 2 multiplied by 100 is 200, and 200 multiplied by 9 is 1,800. The third part, 9², is simply 81. Now, we have three values: 10,000, 1,800, and 81. To find the value of (109)², we simply add these values together: 10,000 + 1,800 + 81. This addition is relatively easy and can be done mentally or with minimal effort on paper. The sum of these numbers will give us the final answer, which is the value of (109)². This step-by-step approach, guided by the algebraic identity, transforms a potentially cumbersome calculation into a series of simple arithmetic operations, highlighting the power and efficiency of algebraic methods.

Step-by-Step Calculation

To further clarify the process, let's break down the calculation of (109)² step-by-step:

  1. Express 109 as a sum: 109 = 100 + 9
  2. Apply the (a + b)² identity: (100 + 9)² = 100² + 2(100)(9) + 9²
  3. Calculate the squares and the product:
    • 100² = 10,000
    • 2(100)(9) = 1,800
    • 9² = 81
  4. Add the results: 10,000 + 1,800 + 81

Adding these values together, we get: 10,000 + 1,800 = 11,800, and then 11,800 + 81 = 11,881. Therefore, (109)² = 11,881. This step-by-step breakdown demonstrates the simplicity and elegance of using algebraic identities to solve mathematical problems. Each step is clear and manageable, reducing the likelihood of errors and making the calculation process more transparent. This method not only provides the correct answer but also enhances understanding of the underlying mathematical principles. By breaking down a complex calculation into smaller, more manageable parts, we can tackle even challenging problems with confidence and accuracy. This approach is a testament to the power of structured problem-solving and the effectiveness of applying the right tools for the job.

Final Result and Conclusion

In conclusion, by applying the (a + b)² identity, we have successfully evaluated (109)². The final result is (109)² = 11,881. This method showcases the power of algebraic identities in simplifying complex calculations. Instead of directly squaring 109, which might be cumbersome, we broke it down into smaller, more manageable parts using the identity. This approach not only makes the calculation easier but also provides a deeper understanding of algebraic principles and their applications. The ability to recognize and apply suitable identities is a valuable skill in mathematics, allowing for efficient and accurate problem-solving. Whether you're dealing with numbers, algebraic expressions, or more complex mathematical concepts, a solid understanding of identities can greatly enhance your mathematical toolkit. This particular example illustrates how a seemingly complex problem can be elegantly solved with the right strategy and the appropriate algebraic tools. By mastering such techniques, you can approach mathematical challenges with greater confidence and proficiency.

  • Identify Suitable Identities: Recognizing which identity to apply is crucial. For numbers close to a multiple of 10 or 100, the (a + b)² or (a - b)² identities are often the most effective.
  • Break Down the Number: Express the number as a sum or difference of two simpler numbers. This makes it easier to apply the chosen identity.
  • Apply the Identity: Substitute the values into the identity and perform the necessary calculations (squaring and multiplication).
  • Simplify and Calculate: Add or subtract the results to arrive at the final answer.
  • Check Your Work: Always double-check your calculations to ensure accuracy.

By following these steps, you can confidently evaluate squares of numbers using algebraic identities, making complex calculations more manageable and efficient. This approach not only aids in solving mathematical problems but also deepens your understanding of algebraic principles.