Simplifying The Quotient Of Radicals A Step-by-Step Guide

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In mathematics, simplifying expressions involving radicals often requires a strategic approach. This article dives deep into the process of rationalizing the denominator to simplify the quotient (6+11)/(5+3)(\sqrt{6}+\sqrt{11})/(\sqrt{5}+\sqrt{3}). We will explore the step-by-step method, underlying principles, and various techniques to tackle such problems effectively. This comprehensive guide aims to provide a clear understanding of how to handle complex quotients and enhance your problem-solving skills in algebra and beyond.

1. Introduction to the Quotient

When dealing with the quotient (6+11)/(5+3)(\sqrt{6}+\sqrt{11})/(\sqrt{5}+\sqrt{3}), our primary goal is to simplify it into a more manageable form. The presence of square roots in the denominator makes the expression somewhat complex. To simplify, we aim to rationalize the denominator, which means eliminating the square roots from the denominator. This is typically achieved by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of (5+3)(\sqrt{5}+\sqrt{3}) is (5−3)(\sqrt{5}-\sqrt{3}). This method utilizes the difference of squares identity, which states that (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2. By applying this, we transform the denominator into a rational number, making the entire expression simpler to handle. Simplifying such expressions is a fundamental skill in algebra and is crucial for solving more complex mathematical problems. Understanding the underlying principles and techniques will enable you to tackle similar challenges with confidence and accuracy. This introductory section sets the stage for a detailed exploration of the simplification process, ensuring a solid foundation for further mathematical endeavors. The subsequent sections will delve into the step-by-step method of rationalizing the denominator and simplifying the quotient, providing a comprehensive guide to mastering this technique.

2. Rationalizing the Denominator: A Step-by-Step Approach

To rationalize the denominator of the quotient (6+11)/(5+3)(\sqrt{6}+\sqrt{11})/(\sqrt{5}+\sqrt{3}), we need to eliminate the square roots from the denominator. The first step involves identifying the conjugate of the denominator, which is (5−3)(\sqrt{5}-\sqrt{3}). We then multiply both the numerator and the denominator by this conjugate. This process ensures that we are effectively multiplying the expression by 1, thus not changing its value, but altering its form. The multiplication can be expressed as follows:

6+115+3×5−35−3\frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}} \times \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}}

Next, we expand both the numerator and the denominator. For the numerator, we use the distributive property (FOIL method):

(6+11)(5−3)=65−63+115−113)(\sqrt{6}+\sqrt{11})(\sqrt{5}-\sqrt{3}) = \sqrt{6}\sqrt{5} - \sqrt{6}\sqrt{3} + \sqrt{11}\sqrt{5} - \sqrt{11}\sqrt{3})

This simplifies to:

30−18+55−33\sqrt{30} - \sqrt{18} + \sqrt{55} - \sqrt{33}

For the denominator, we use the difference of squares identity: (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2. So,

(5+3)(5−3)=(5)2−(3)2=5−3=2(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3}) = (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2

Now, we combine the simplified numerator and denominator:

30−18+55−332\frac{\sqrt{30} - \sqrt{18} + \sqrt{55} - \sqrt{33}}{2}

We can further simplify 18\sqrt{18} as 9×2=32\sqrt{9 \times 2} = 3\sqrt{2}. Thus, the expression becomes:

30−32+55−332\frac{\sqrt{30} - 3\sqrt{2} + \sqrt{55} - \sqrt{33}}{2}

This final expression represents the simplified form of the original quotient, with the denominator rationalized. This step-by-step approach ensures clarity and accuracy in handling complex expressions involving radicals. Rationalizing the denominator is a critical technique in algebra, and mastering this process is essential for advanced mathematical problem-solving.

3. Expanding the Numerator: Detailed Breakdown

Expanding the numerator in the expression (6+11)(5−3)(\sqrt{6}+\sqrt{11})(\sqrt{5}-\sqrt{3}) requires a careful application of the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This method ensures that each term in the first binomial is multiplied by each term in the second binomial. Let’s break down the process step by step.

First, we multiply the First terms: 6×5\sqrt{6} \times \sqrt{5}. This yields 6×5=30\sqrt{6 \times 5} = \sqrt{30}.

Next, we multiply the Outer terms: 6×(−3)\sqrt{6} \times (-\sqrt{3}). This results in −6×3=−18-\sqrt{6 \times 3} = -\sqrt{18}.

Then, we multiply the Inner terms: 11×5\sqrt{11} \times \sqrt{5}. This gives us 11×5=55\sqrt{11 \times 5} = \sqrt{55}.

Finally, we multiply the Last terms: 11×(−3)\sqrt{11} \times (-\sqrt{3}). This equals −11×3=−33-\sqrt{11 \times 3} = -\sqrt{33}.

Combining these terms, we get:

30−18+55−33\sqrt{30} - \sqrt{18} + \sqrt{55} - \sqrt{33}

Now, we look for further simplifications. We can simplify 18\sqrt{18} by factoring out the largest perfect square, which is 9. So, 18=9×2=9×2=32\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}.

The simplified numerator then becomes:

30−32+55−33\sqrt{30} - 3\sqrt{2} + \sqrt{55} - \sqrt{33}

This detailed breakdown illustrates how the distributive property is applied to expand and simplify the numerator. Each step is crucial to ensure accuracy and to handle complex expressions involving radicals effectively. Mastering this technique is essential for simplifying algebraic expressions and solving mathematical problems involving square roots. The ability to expand and simplify expressions accurately is a cornerstone of algebraic proficiency.

4. Simplifying the Denominator: Utilizing the Difference of Squares

Simplifying the denominator of the quotient involves a strategic application of the difference of squares identity. In our case, the denominator is (5+3)(5−3)(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3}). The difference of squares identity states that (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2. This identity is a powerful tool for rationalizing denominators, as it eliminates square roots by squaring them. Applying this identity to our denominator, we can clearly see how it simplifies the expression.

Let's identify aa and bb in our expression. Here, a=5a = \sqrt{5} and b=3b = \sqrt{3}. Substituting these values into the difference of squares identity, we get:

(5+3)(5−3)=(5)2−(3)2(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3}) = (\sqrt{5})^2 - (\sqrt{3})^2

Now, we square each term:

(5)2=5(\sqrt{5})^2 = 5

(3)2=3(\sqrt{3})^2 = 3

So, the expression becomes:

5−3=25 - 3 = 2

Therefore, the denominator simplifies to 2. This is a significant step in rationalizing the denominator, as we have successfully eliminated the square roots from the denominator. The difference of squares identity is a fundamental concept in algebra and is frequently used in simplifying expressions and solving equations. Understanding and applying this identity correctly is crucial for algebraic manipulation and problem-solving. This simplification makes the entire quotient easier to handle and understand, showcasing the power and efficiency of algebraic identities in mathematical simplifications. The ability to quickly and accurately simplify expressions using identities is a hallmark of strong mathematical skills.

5. Combining the Simplified Numerator and Denominator

After simplifying both the numerator and the denominator, the next crucial step is to combine them to form the simplified quotient. From our previous calculations, we have the simplified numerator as 30−32+55−33\sqrt{30} - 3\sqrt{2} + \sqrt{55} - \sqrt{33} and the simplified denominator as 2. Now, we express the quotient as:

30−32+55−332\frac{\sqrt{30} - 3\sqrt{2} + \sqrt{55} - \sqrt{33}}{2}

This expression represents the quotient with a rationalized denominator. It is essential to check if any further simplifications are possible. In this case, none of the square roots in the numerator can be simplified further, and there are no common factors between the numerator and the denominator. Therefore, this is the final simplified form of the original quotient. This process of combining the simplified parts highlights the importance of each individual step in the overall simplification process. Ensuring that both the numerator and the denominator are in their simplest forms before combining them is key to achieving the most simplified expression. This step also emphasizes the clarity and elegance that comes with simplifying complex mathematical expressions. The final expression is not only mathematically accurate but also easier to interpret and work with in further calculations. The ability to systematically simplify and combine expressions is a fundamental skill in mathematics, paving the way for tackling more complex problems and concepts.

6. Conclusion: Mastering Quotient Simplification

In conclusion, simplifying the quotient (6+11)/(5+3)(\sqrt{6}+\sqrt{11})/(\sqrt{5}+\sqrt{3}) involves a series of strategic steps, primarily focused on rationalizing the denominator. This process not only simplifies the expression but also makes it easier to handle in further mathematical operations. The key techniques we employed include identifying the conjugate of the denominator, multiplying both the numerator and denominator by this conjugate, expanding the products, and simplifying the resulting expression.

We began by recognizing the need to eliminate the square roots from the denominator. This led us to the concept of the conjugate, which, when multiplied by the original denominator, eliminates the square roots through the difference of squares identity. The step-by-step approach ensured that each term was accounted for, and no algebraic rule was violated. The distributive property, also known as the FOIL method, played a crucial role in expanding the products in the numerator, while the difference of squares identity efficiently simplified the denominator.

The simplified quotient, 30−32+55−332\frac{\sqrt{30} - 3\sqrt{2} + \sqrt{55} - \sqrt{33}}{2}, is the culmination of these efforts. It represents the original expression in a more manageable form, with a rationalized denominator. This process underscores the importance of algebraic manipulation and the strategic application of mathematical identities.

Mastering these techniques is not just about solving this specific problem; it's about building a solid foundation in algebra. These skills are transferable and will be invaluable in tackling more complex mathematical challenges. The ability to simplify expressions, rationalize denominators, and apply algebraic identities is essential for success in higher-level mathematics, including calculus, linear algebra, and beyond. Therefore, understanding and practicing these techniques is a worthwhile investment in your mathematical journey. This comprehensive guide aims to empower you with the knowledge and skills necessary to approach similar problems with confidence and accuracy, ensuring a deeper understanding of mathematical principles and their applications.