Simplifying The Quotient (8 + √2) / (4 - √7) A Step-by-Step Guide
Navigating the realm of algebraic expressions often involves simplifying complex quotients. In this comprehensive guide, we'll embark on a step-by-step journey to unravel the quotient of (8 + √2 divided by 4 - √7). Our mission is to transform this seemingly intricate expression into its most simplified and elegant form. To achieve this, we'll employ a powerful technique known as rationalizing the denominator. This method allows us to eliminate the radical from the denominator, paving the way for a clearer and more manageable expression. So, let's dive in and unlock the secrets of this quotient!
The Art of Rationalizing the Denominator
At the heart of our simplification strategy lies the concept of rationalizing the denominator. This technique empowers us to banish radicals from the denominator of a fraction, resulting in a more refined and presentable expression. The key to rationalization lies in the ingenious use of the conjugate. The conjugate of a binomial expression, such as our denominator (4 - √7), is formed by simply changing the sign between the terms. Thus, the conjugate of (4 - √7) is (4 + √7).
To rationalize the denominator, we'll multiply both the numerator and denominator of our original quotient by the conjugate of the denominator. This crucial step ensures that we maintain the value of the expression while simultaneously eliminating the radical in the denominator. Let's witness this transformation in action:
Original Quotient:
(8 + √2) / (4 - √7)
Multiply by Conjugate:
[(8 + √2) * (4 + √7)] / [(4 - √7) * (4 + √7)]
By multiplying both the numerator and denominator by the conjugate, we've set the stage for a remarkable simplification. The denominator, once burdened by a radical, is now poised to transform into a rational number.
Unleashing the Power of Multiplication
With the conjugate in place, our next step involves meticulously multiplying the binomials in both the numerator and denominator. This process, often referred to as FOIL (First, Outer, Inner, Last), ensures that each term in the first binomial interacts with every term in the second binomial. Let's embark on this multiplication journey:
Numerator Multiplication:
(8 + √2) * (4 + √7) = (8 * 4) + (8 * √7) + (√2 * 4) + (√2 * √7)
Simplifying, we get:
32 + 8√7 + 4√2 + √14
Denominator Multiplication:
(4 - √7) * (4 + √7) = (4 * 4) + (4 * √7) + (-√7 * 4) + (-√7 * √7)
Simplifying, we get:
16 + 4√7 - 4√7 - 7
Notice how the terms involving √7 neatly cancel each other out, leaving us with:
16 - 7 = 9
The denominator has gracefully transformed into the rational number 9, a testament to the power of rationalization.
The Simplified Quotient Emerges
With the multiplication complete, we now have a clearer picture of our quotient:
[32 + 8√7 + 4√2 + √14] / 9
This expression, while significantly simpler than our original quotient, can be further refined. We'll rearrange the terms in the numerator to present the expression in a more conventional form:
(32 + 8√7 + 4√2 + √14) / 9
This is the simplified form of our quotient. Let's examine the answer choices provided and identify the one that matches our result.
Evaluating the Answer Choices
Now, let's carefully compare our simplified quotient with the answer choices provided:
A. (9√7 + √14) / -3 B. (38 - 9√7 + 4√2 - √14) / 9 C. (38 + 9√7 + 4√2 + √14) / 9 D. 18 / 8
By meticulous comparison, we can confidently identify the correct answer choice.
Upon close inspection, we find that option C, (32 + 8√7 + 4√2 + √14) / 9, perfectly matches our simplified quotient.
Therefore, the correct answer is C.
Key Takeaways and Insights
Through this exploration, we've not only successfully simplified the quotient but also gained valuable insights into the art of rationalizing denominators and simplifying algebraic expressions. Let's recap the key takeaways:
- Rationalizing the denominator is a powerful technique for eliminating radicals from the denominator of a fraction.
- The conjugate of a binomial expression is formed by changing the sign between the terms.
- Multiplying both the numerator and denominator by the conjugate is the cornerstone of rationalization.
- The FOIL method (First, Outer, Inner, Last) is essential for multiplying binomials accurately.
- Careful simplification and comparison are crucial for arriving at the final answer.
By mastering these concepts and techniques, you'll be well-equipped to tackle a wide range of algebraic challenges. Keep practicing, keep exploring, and watch your mathematical prowess flourish!
Further Exploration and Practice
To solidify your understanding and hone your skills, consider exploring additional examples and practice problems related to rationalizing denominators and simplifying algebraic expressions. Numerous resources, including textbooks, online tutorials, and practice websites, are readily available to support your learning journey. Embrace the challenge, and you'll be amazed at the progress you make!
In conclusion, we've successfully unraveled the quotient of (8 + √2 divided by 4 - √7) and identified the correct answer choice. Through the power of rationalization, meticulous multiplication, and careful simplification, we've transformed a seemingly complex expression into its elegant, simplified form. Remember, the world of algebra is full of fascinating challenges and rewarding discoveries. Keep exploring, keep learning, and keep unlocking the beauty of mathematics!
