How To Rationalize The Denominator Of (3 - 5√2) / (2 - 7√5)

Rationalizing the denominator is a fundamental technique in algebra used to eliminate radicals from the denominator of a fraction. This process simplifies expressions, making them easier to work with in further calculations and analyses. In this comprehensive guide, we will walk through the steps to rationalize the denominator of the expression $rac{3-5 \sqrt{2}}{2-7 \sqrt{5}}$, providing a detailed explanation and insights into the underlying mathematical principles.

Understanding the Need for Rationalization

Before diving into the steps, it's essential to understand why rationalizing the denominator is important. In mathematics, it's generally preferred to have rational numbers (integers or fractions of integers) in the denominator of a fraction. This convention stems from several reasons:

• Simplification: Rationalizing the denominator simplifies the expression, making it easier to compare, add, subtract, multiply, and divide fractions.
• Standard Form: It presents the expression in a standard form, facilitating communication and consistency across mathematical contexts.
• Ease of Calculation: Calculations involving fractions with rational denominators are often easier to perform manually or with calculators.

Rationalizing the denominator involves multiplying both the numerator and the denominator of a fraction by a specific value that will eliminate the radical in the denominator. This value is typically the conjugate of the denominator. By understanding the significance of this process, we can appreciate its role in mathematical simplification and problem-solving.

Identifying the Conjugate

The first step in rationalizing the denominator is to identify the conjugate of the denominator. The conjugate of a binomial expression in the form a + b√c is a - b√c, and vice versa. The key idea behind using the conjugate is that when you multiply a binomial expression by its conjugate, the radical term is eliminated due to the difference of squares identity: (a + b)(a - b) = a² - b². Therefore, the conjugate of the denominator plays a critical role in the rationalization process.

In our expression, the denominator is 2 - 7√5. The conjugate of 2 - 7√5 is 2 + 7√5. This means we will multiply both the numerator and the denominator of the fraction by 2 + 7√5 to rationalize the denominator. Identifying the conjugate correctly is crucial because it sets the stage for eliminating the radical term in the denominator. Understanding this concept is fundamental to mastering rationalization techniques in algebra.

Multiplying by the Conjugate

Once we've identified the conjugate, the next step is to multiply both the numerator and the denominator of the original fraction by this conjugate. This process is based on the fundamental principle that multiplying a fraction by a form of 1 (in this case, the conjugate divided by itself) does not change its value. It only changes its form. The multiplication step is where the algebraic manipulation leads to the elimination of the radical in the denominator.

So, we multiply both the numerator (3 - 5√2) and the denominator (2 - 7√5) by the conjugate (2 + 7√5). This gives us:

$$\frac{3-5 \sqrt{2}}{2-7 \sqrt{5}} \times \frac{2+7 \sqrt{5}}{2+7 \sqrt{5}}$$

This multiplication sets up the subsequent steps, where we'll expand the products in the numerator and the denominator. The denominator will simplify nicely due to the difference of squares, which is the primary reason for using the conjugate. This step is crucial for setting up the simplification process and arriving at the rationalized form of the fraction. Remember, multiplying by the conjugate is a strategic move to leverage the difference of squares identity and remove the radical from the denominator.

Expanding the Numerator and Denominator

After multiplying by the conjugate, the next crucial step is to expand both the numerator and the denominator. This involves applying the distributive property (also known as the FOIL method) to multiply the binomials. The expansion process is where we transform the products into sums and differences of terms, setting the stage for simplification. Accurate expansion is essential to avoid errors and ensure the correct final result.

Let's expand the numerator:

(3 - 5√2)(2 + 7√5) = 3(2) + 3(7√5) - 5√2(2) - 5√2(7√5)

This simplifies to:

6 + 21√5 - 10√2 - 35√(10)

Now, let's expand the denominator:

(2 - 7√5)(2 + 7√5) = 2(2) + 2(7√5) - 7√5(2) - 7√5(7√5)

This simplifies to:

4 + 14√5 - 14√5 - 49(5)

The expansion process reveals the terms that will combine and simplify, especially in the denominator where the radical terms should cancel out, leaving us with a rational number. This step bridges the gap between multiplication and simplification, and it's vital for achieving the rationalized form of the fraction. Careless expansion can lead to incorrect results, so attention to detail is key.

Simplifying the Expression

Following the expansion, the next critical step is to simplify the expression. This involves combining like terms in both the numerator and the denominator. The simplification process is crucial for reducing the expression to its simplest form, making it easier to interpret and use in subsequent calculations. Accurate simplification ensures that the final result is clear and concise.

Looking at the expanded numerator, we have:

6 + 21√5 - 10√2 - 35√(10)

There are no like terms to combine here, so the numerator remains as is.

Now, let's simplify the denominator. We have:

4 + 14√5 - 14√5 - 49(5)

The terms 14√5 and -14√5 cancel each other out. Also, 49(5) = 245. So, the denominator simplifies to:

4 - 245 = -241

Thus, the expression becomes:

$$\frac{6 + 21\sqrt{5} - 10\sqrt{2} - 35\sqrt{10}}{-241}$$

The simplification step not only reduces the expression but also brings us closer to the final rationalized form. By combining like terms and performing basic arithmetic, we make the expression more manageable and easier to work with. This step is essential for presenting the solution in its most understandable and practical form. Careful simplification is the key to avoiding errors and arriving at the correct answer.

Final Rationalized Form

After simplifying the expression, we arrive at the final rationalized form. This involves ensuring that there are no radicals in the denominator and presenting the expression in its simplest terms. The final form represents the culmination of all the previous steps, providing a clear and concise solution. Reaching the final form demonstrates a thorough understanding of the rationalization process and its application.

From the previous step, we have:

$$\frac{6 + 21\sqrt{5} - 10\sqrt{2} - 35\sqrt{10}}{-241}$$

To present the expression in a more conventional form, we can distribute the negative sign from the denominator to the numerator:

$$\frac{-(6 + 21\sqrt{5} - 10\sqrt{2} - 35\sqrt{10})}{241}$$

This can be further written as:

$$\frac{-6 - 21\sqrt{5} + 10\sqrt{2} + 35\sqrt{10}}{241}$$

Or, rearranging the terms, we have:

$$\frac{35\sqrt{10} + 10\sqrt{2} - 21\sqrt{5} - 6}{241}$$

This is the final rationalized form of the given expression. The denominator is now a rational number, and the expression is simplified as much as possible. The final form showcases the effectiveness of the rationalization process in transforming a complex expression into a more manageable one. Reviewing each step and ensuring accurate execution is crucial for achieving the correct final form.

Conclusion

Rationalizing the denominator is a crucial skill in algebra that simplifies expressions and makes them easier to work with. By understanding the process of identifying the conjugate, multiplying by it, expanding the terms, and simplifying the expression, we can effectively eliminate radicals from the denominator. In the case of $\frac3-5 \sqrt{2}}{2-7 \sqrt{5}}$, we have successfully rationalized the denominator to obtain the final form $\frac{35\sqrt{10 + 10\sqrt{2} - 21\sqrt{5} - 6}{241}$. Mastering this technique enhances problem-solving capabilities and provides a solid foundation for more advanced mathematical concepts.