Rationalizing Denominators Multiplying By The Right Fraction

Rationalizing denominators is a fundamental skill in mathematics, particularly when dealing with fractions that contain radicals (square roots, cube roots, etc.) in the denominator. The goal is to manipulate the fraction so that the denominator becomes a rational number, meaning it can be expressed as a simple fraction or integer. This process often simplifies expressions, makes them easier to work with, and aligns with standard mathematical conventions.

To effectively rationalize a denominator, you need to identify the appropriate fraction to multiply by. This fraction should be equivalent to 1, ensuring that you're not changing the value of the original expression, only its form. The key is to choose a fraction that will eliminate the radical from the denominator when multiplied. In this comprehensive guide, we'll delve into the specific case of rationalizing the denominator of the fraction $\frac{3}{\sqrt{17}-\sqrt{2}}$. We'll explore the underlying principles, the step-by-step process, and the reasoning behind each action, making sure you understand not just how to do it, but why it works. We'll also touch upon the broader applications of rationalizing denominators in mathematics and beyond.

Understanding the Problem: The Irrational Denominator

Our starting point is the fraction $\frac{3}{\sqrt{17}-\sqrt{2}}$. The denominator, $\sqrt{17}-\sqrt{2}$, is an irrational number because it involves the difference of two square roots that are not perfect squares. This means the denominator cannot be expressed as a simple fraction or integer. Having an irrational denominator can make further calculations and simplifications cumbersome. For instance, if we needed to add this fraction to another fraction, the irrational denominator would complicate the process of finding a common denominator. Moreover, in many mathematical contexts, it's considered standard practice to express fractions with rational denominators.

To address this issue, we employ the technique of rationalizing the denominator. The fundamental idea behind this technique is to multiply the given fraction by a carefully chosen form of 1. This allows us to change the appearance of the fraction without altering its numerical value. The specific form of 1 we choose is crucial; it must be constructed in a way that eliminates the radical from the denominator when multiplied. This usually involves using the conjugate of the denominator, which we'll discuss in detail in the next section. By understanding the nature of irrational denominators and the rationale behind rationalization, we set the stage for successfully tackling the problem at hand.

The Conjugate: Our Key to Rationalization

The most effective way to rationalize a denominator containing a difference or sum of square roots is by multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate is formed by simply changing the sign between the terms in the original denominator. In our case, the denominator is $\sqrt{17}-\sqrt{2}$, so its conjugate is $\sqrt{17}+\sqrt{2}$. The magic of using the conjugate lies in the difference of squares pattern: $(a - b)(a + b) = a^2 - b^2$. When we multiply the denominator by its conjugate, the square roots will be squared, effectively eliminating them and leaving us with rational numbers.

Consider what happens when we multiply $(\sqrt{17}-\sqrt{2})$ by its conjugate $(\sqrt{17}+\sqrt{2})$:

$(\sqrt{17}-\sqrt{2})(\sqrt{17}+\sqrt{2}) = (\sqrt{17})^2 - (\sqrt{2})^2 = 17 - 2 = 15$

As you can see, the result is the rational number 15. This illustrates the power of the conjugate in eliminating square roots from the denominator. By multiplying both the numerator and the denominator of our original fraction by this conjugate, we maintain the fraction's value while transforming its denominator into a rational number. This is a crucial step in simplifying expressions and making them easier to manipulate in further calculations. In the next section, we'll apply this principle to the given fraction and see how it works in practice.

Step-by-Step Solution: Multiplying by the Conjugate

Now that we understand the concept of the conjugate and its role in rationalizing denominators, let's apply this knowledge to the fraction $\frac{3}{\sqrt{17}-\sqrt{2}}$. The first step is to identify the conjugate of the denominator, which, as we established, is $\sqrt{17}+\sqrt{2}$. To rationalize the denominator, we will multiply both the numerator and the denominator of the fraction by this conjugate. This is equivalent to multiplying the fraction by 1, so we are not changing its value.

The multiplication looks like this:

$\frac{3}{\sqrt{17}-\sqrt{2}} \times \frac{\sqrt{17}+\sqrt{2}}{\sqrt{17}+\sqrt{2}}$

Multiplying the numerators, we get:

$3(\sqrt{17}+\sqrt{2}) = 3\sqrt{17} + 3\sqrt{2}$

As we saw earlier, multiplying the denominators gives us:

$(\sqrt{17}-\sqrt{2})(\sqrt{17}+\sqrt{2}) = 17 - 2 = 15$

So, our fraction now becomes:

$\frac{3\sqrt{17} + 3\sqrt{2}}{15}$

We can simplify this fraction further by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

$\frac{3\sqrt{17} + 3\sqrt{2}}{15} = \frac{\sqrt{17} + \sqrt{2}}{5}$

Thus, the equivalent fraction with a rational denominator is $\frac{\sqrt{17} + \sqrt{2}}{5}$. This step-by-step process demonstrates how multiplying by the conjugate effectively eliminates the radicals from the denominator, resulting in a simplified and more mathematically convenient form of the original fraction. In the next section, we'll summarize the key steps and discuss the broader implications of this technique.

Key Steps Summarized: A Recap of the Process

To rationalize the denominator of a fraction like $\frac{3}{\sqrt{17}-\sqrt{2}}$, we follow a few key steps. These steps ensure that we eliminate the radical from the denominator while maintaining the fraction's original value. Let's recap the process:

1. Identify the denominator: In our case, the denominator is $\sqrt{17}-\sqrt{2}$.

2. Find the conjugate: The conjugate is formed by changing the sign between the terms in the denominator. The conjugate of $\sqrt{17}-\sqrt{2}$ is $\sqrt{17}+\sqrt{2}$.

3. Multiply by a form of 1: Multiply both the numerator and the denominator of the original fraction by the conjugate. This is equivalent to multiplying by 1, which doesn't change the fraction's value:

   $\frac{3}{\sqrt{17}-\sqrt{2}} \times \frac{\sqrt{17}+\sqrt{2}}{\sqrt{17}+\sqrt{2}}$

4. Multiply out the numerators: $3(\sqrt{17}+\sqrt{2}) = 3\sqrt{17} + 3\sqrt{2}$

5. Multiply out the denominators: $(\sqrt{17}-\sqrt{2})(\sqrt{17}+\sqrt{2}) = 17 - 2 = 15$

6. Simplify the resulting fraction: $\frac{3\sqrt{17} + 3\sqrt{2}}{15} = \frac{\sqrt{17} + \sqrt{2}}{5}$

By following these steps, we successfully transformed the original fraction into an equivalent fraction with a rational denominator. This process not only simplifies the fraction but also makes it easier to work with in further mathematical operations. In the final section, we'll discuss the broader significance of rationalizing denominators and its applications in various mathematical contexts.

The Significance of Rationalizing Denominators: Beyond Simplification

While rationalizing the denominator is a valuable simplification technique, its significance extends beyond merely making fractions look neater. It plays a crucial role in several areas of mathematics and related fields. One primary reason for rationalizing denominators is to facilitate further calculations. When dealing with complex expressions involving fractions, having a rational denominator makes it significantly easier to add, subtract, multiply, or divide fractions. It simplifies the process of finding common denominators and combining terms.

Moreover, rationalizing denominators is essential in standardizing mathematical notation. It's a convention in mathematics to express fractions with rational denominators whenever possible. This ensures consistency and clarity in mathematical communication. It also makes it easier to compare different expressions and recognize equivalent forms.

In advanced mathematics, such as calculus and analysis, rationalizing denominators can be a crucial step in evaluating limits, derivatives, and integrals. It can transform seemingly intractable expressions into forms that are more amenable to these operations. For instance, rationalizing the denominator might allow you to remove a discontinuity or simplify an integrand.

Beyond pure mathematics, the technique of rationalizing denominators finds applications in fields like physics, engineering, and computer science, where mathematical expressions involving radicals often arise. In these contexts, rationalizing denominators can simplify calculations and make results easier to interpret.

In conclusion, rationalizing the denominator is not just a cosmetic procedure; it's a fundamental technique that enhances mathematical clarity, facilitates calculations, and has wide-ranging applications across various disciplines. Mastering this skill is essential for anyone pursuing further studies in mathematics or related fields.

Answer

The fraction that will produce an equivalent fraction with a rational denominator when multiplied by $\frac{3}{\sqrt{17}-\sqrt{2}}$ is:

B. $\frac{\sqrt{17}+\sqrt{2}}{\sqrt{17}+\sqrt{2}}$