Prove 1 10 Less Than Square Root 101 Minus Square Root 99 Without Calculator

In this article, we will explore a fascinating inequality problem that challenges our understanding of radicals and algebraic manipulation. The problem asks us to show that $\frac{1}{10} < \sqrt{101} - \sqrt{99}$ without the aid of a calculator. This seemingly simple problem requires a clever approach, utilizing techniques such as radical conjugates and careful estimation. It's a great exercise in honing our algebraic skills and understanding the behavior of square roots. This problem often arises in intermediate algebra courses and serves as a good example of how to work with inequalities involving radicals. We'll break down the solution step-by-step, providing a clear and concise explanation of each step involved. Solving this problem not only helps us understand the specific inequality but also equips us with valuable problem-solving strategies applicable to a broader range of mathematical challenges. So, let's embark on this mathematical journey and discover the elegant solution to this intriguing inequality.

Understanding the Problem

Before diving into the solution, it's crucial to understand what the problem is asking. We need to prove that the difference between the square roots of 101 and 99 is greater than 1/10. The challenge lies in doing this without relying on a calculator to approximate the square roots. This means we need to find a way to manipulate the expression algebraically to reveal the inequality. Our understanding of algebraic manipulations, especially those involving radicals and inequalities, will be pivotal in solving this problem. We need to think about how we can get rid of the square roots in the denominator (if we introduce one) or how we can make the comparison easier without actually computing the square roots. This involves recalling and applying properties of inequalities and radicals learned in algebra. This preliminary understanding is key to formulating a strategy for tackling the problem effectively. The goal here is not just to find an answer, but to rigorously demonstrate the inequality using mathematical principles.

The Radical Conjugate Approach

The most effective approach to tackling this inequality involves using the concept of radical conjugates. The conjugate of $\sqrt{101} - \sqrt{99}$ is $\sqrt{101} + \sqrt{99}$. Multiplying an expression by its conjugate is a common technique used to rationalize denominators or, in this case, to eliminate the square roots. When we multiply $(\sqrt{101} - \sqrt{99})$ by $(\sqrt{101} + \sqrt{99})$, we are essentially using the difference of squares identity: $(a - b)(a + b) = a^2 - b^2$. This identity is a cornerstone of algebraic manipulation and is particularly useful when dealing with radicals. Applying this to our problem, we get $(\sqrt{101})^2 - (\sqrt{99})^2 = 101 - 99 = 2$. This simple result is the key to unlocking the solution. By multiplying the original expression by its conjugate, we transform it into a form that is easier to compare with 1/10. This strategy highlights the power of algebraic manipulation in simplifying complex expressions and revealing hidden relationships. The use of conjugates is not just a trick; it's a systematic way of dealing with expressions involving radicals, and mastering this technique is essential for solving a wide range of algebraic problems.

Applying the Conjugate

To apply the radical conjugate, we'll multiply both sides of the inequality we want to prove by the conjugate $(\sqrt{101} + \sqrt{99})$. However, to do this correctly, we need to manipulate the expression $\frac{1}{10} < \sqrt{101} - \sqrt{99}$ first. We can multiply both sides of the inequality by $(\sqrt{101} + \sqrt{99})$, which is a positive number, so the inequality sign remains the same. This gives us: $\frac{1}{10}(\sqrt{101} + \sqrt{99}) < (\sqrt{101} - \sqrt{99})(\sqrt{101} + \sqrt{99})$. Now, we simplify the right side using the difference of squares identity, as discussed earlier. This simplifies to $101 - 99 = 2$. So, the inequality now looks like: $\frac{1}{10}(\sqrt{101} + \sqrt{99}) < 2$. This step is crucial because it transforms the original inequality into a more manageable form. We've eliminated the subtraction of square roots on one side, making it easier to isolate and compare terms. This highlights the importance of strategically applying algebraic operations to simplify expressions and reveal underlying relationships. The next step involves further manipulation to isolate the square roots and make a clear comparison.

Simplifying the Inequality

Now we have the inequality $\frac{1}{10}(\sqrt{101} + \sqrt{99}) < 2$. To further simplify, we can multiply both sides of the inequality by 10, which again doesn't change the direction of the inequality since 10 is positive. This gives us: $\sqrt{101} + \sqrt{99} < 20$. Now, we need to show that this inequality holds true. To do this, we can think about the values of $\sqrt{101}$ and $\sqrt{99}$. We know that $\sqrt{100} = 10$. Since 101 is slightly greater than 100, $\sqrt{101}$ will be slightly greater than 10. Similarly, since 99 is slightly less than 100, $\sqrt{99}$ will be slightly less than 10. This understanding of the approximate values of square roots is essential for making comparisons without a calculator. By relating the square roots to a perfect square (100), we can establish bounds for their values. This step demonstrates the importance of number sense and the ability to estimate values effectively. The next step involves using these estimations to prove the inequality.

Estimating the Square Roots

We know that $\sqrt{101}$ is slightly greater than 10 and $\sqrt{99}$ is slightly less than 10. To be more precise, we can consider that $10^2 = 100$ and $10.1^2 = 102.01$. This tells us that $\sqrt{101}$ is less than 10.1. Similarly, we can consider that $9.9^2 = 98.01$, which means $\sqrt{99}$ is greater than 9.9. Therefore, we have the following bounds: $\sqrt{101} < 10.1$ and $\sqrt{99} > 9.9$. Adding these two inequalities, we get: $\sqrt{101} + \sqrt{99} < 10.1 + 10 = 20.1$ and $\sqrt{101} + \sqrt{99} > 9.9 + 10 = 19.9$. This estimation is crucial because it provides us with a range within which the sum of the square roots must lie. By establishing these bounds, we can confidently compare the sum to 20 and demonstrate the validity of the inequality. This step highlights the importance of careful estimation and the use of nearby perfect squares to approximate the values of square roots. The final step involves using these bounds to definitively prove the original inequality.

Proving the Inequality

From our estimations, we have $\sqrt{101} < 10.05$ (a tighter bound) and $\sqrt{99} < 10$. Therefore, $\sqrt{101} + \sqrt{99} < 10.05 + 9.95 = 20$. This confirms our earlier inequality: $\sqrt{101} + \sqrt{99} < 20$. Since we arrived at this inequality by performing valid algebraic manipulations on the original inequality $\frac{1}{10} < \sqrt{101} - \sqrt{99}$, we have successfully proven that the original inequality holds true. This final step brings together all the previous steps, demonstrating the logical flow of the solution. We started with the original inequality, applied the radical conjugate technique, simplified the expression, estimated the square roots, and finally, showed that the simplified inequality is indeed true. This comprehensive approach showcases the power of mathematical reasoning and the importance of each step in arriving at a valid conclusion. The successful proof not only solves the problem but also reinforces our understanding of algebraic manipulation, inequalities, and the behavior of radicals.

Conclusion

In conclusion, we have successfully shown that $\frac{1}{10} < \sqrt{101} - \sqrt{99}$ without using a calculator. We achieved this by employing the technique of radical conjugates, simplifying the resulting expression, and carefully estimating the values of the square roots. This problem serves as a great example of how algebraic manipulation and estimation can be used to solve complex inequalities. The key takeaway from this problem is the power of strategic thinking and the application of fundamental algebraic principles. By understanding and utilizing techniques like radical conjugates, we can transform seemingly difficult problems into manageable steps. This problem also reinforces the importance of number sense and the ability to estimate values effectively, especially when dealing with radicals. The solution demonstrates a clear and logical progression, highlighting the importance of each step in arriving at a valid conclusion. Ultimately, mastering these problem-solving strategies not only enhances our mathematical skills but also equips us with valuable tools for tackling a wide range of challenges in various fields.

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FAQ

Q: What is a radical conjugate and why is it used? A: A radical conjugate is formed by changing the sign between two terms in a binomial expression involving radicals (e.g., the conjugate of √a - √b is √a + √b). It's used to rationalize denominators or eliminate square roots in expressions, often simplifying complex algebraic manipulations.

Q: How do you estimate square roots without a calculator? A: You can estimate square roots by identifying the nearest perfect squares. For example, to estimate √101, we know that √100 = 10. Since 101 is slightly more than 100, √101 will be slightly more than 10. We can refine this estimation by considering the squares of numbers close to 10, such as 10.1.

Q: Why is it important to show steps in a mathematical proof? A: Showing steps in a mathematical proof is crucial for clarity, logical flow, and verification. Each step should follow logically from the previous one, ensuring that the argument is sound and the conclusion is valid. It also allows others to follow and understand the reasoning.

Q: Can this method be applied to other inequality problems involving radicals? A: Yes, the method of using radical conjugates can be applied to a variety of inequality problems involving radicals. It is a powerful technique for simplifying expressions and making comparisons easier. The key is to identify the conjugate and apply it strategically to eliminate the radicals.

Q: What are some common mistakes to avoid when solving inequality problems? A: Some common mistakes include forgetting to reverse the inequality sign when multiplying or dividing by a negative number, making incorrect assumptions about the signs of variables, and not properly justifying each step in the solution. It's also important to check the solution to ensure it satisfies the original inequality.

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