Rationalize The Denominator Of 17/(13-sqrt(10)) A Step-by-Step Guide

In mathematics, simplifying expressions often involves eliminating radicals from the denominator. This process, known as rationalizing the denominator, transforms an expression with an irrational denominator into an equivalent one with a rational denominator. This not only makes the expression easier to work with but also adheres to the standard mathematical practice of presenting expressions in their simplest form. This article aims to provide a detailed explanation of rationalizing denominators, focusing on expressions involving square roots, and offer step-by-step guidance with an illustrative example. Specifically, we will address the simplification of the expression $\frac{17}{13-\sqrt{10}}$, demonstrating how to eliminate the square root from the denominator and arrive at a simplified answer. The techniques discussed here are essential for various mathematical operations, including solving equations, simplifying complex fractions, and performing calculus operations. Understanding how to rationalize denominators is a fundamental skill that enhances one's ability to manipulate and interpret mathematical expressions effectively. Mastering this skill ensures accurate and efficient problem-solving in a wide range of mathematical contexts. In the subsequent sections, we will delve into the mechanics of rationalizing denominators, providing clarity and actionable steps to tackle such problems.

Understanding Rationalizing the Denominator

Rationalizing the denominator is a crucial technique in algebra that simplifies expressions by removing radicals, such as square roots, cube roots, or other roots, from the denominator of a fraction. The primary goal is to transform a fraction with an irrational denominator into an equivalent fraction with a rational denominator. This process enhances the clarity and simplicity of mathematical expressions, making them easier to manipulate and interpret. The need for rationalizing denominators arises from the convention in mathematics to express final answers without radicals in the denominator. This convention helps in standardizing the form of mathematical expressions, facilitating easier comparison and further calculations. When dealing with expressions like $\frac{1}{\sqrt{2}}$, the denominator contains a square root, making it irrational. To rationalize this denominator, we multiply both the numerator and the denominator by $\sqrt{2}$, resulting in $\frac{\sqrt{2}}{2}$. This transformation eliminates the square root from the denominator, rendering it a rational number. The process becomes slightly more complex when the denominator is a binomial expression involving a square root, such as $a + \sqrt{b}$ or $a - \sqrt{b}$. In such cases, we use the conjugate of the denominator to rationalize it. The conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$, and vice versa. Multiplying the denominator by its conjugate leverages the difference of squares identity, $(a + b)(a - b) = a^2 - b^2$, which eliminates the square root. For instance, to rationalize a denominator like $13 - \sqrt{10}$, we multiply both the numerator and the denominator by its conjugate, $13 + \sqrt{10}$. This technique is fundamental in simplifying expressions and is widely applied in various areas of mathematics, including algebra, calculus, and complex analysis. By mastering this skill, students can confidently handle complex mathematical problems and ensure that their solutions are presented in the most simplified and conventional form.

Step-by-Step Guide to Rationalizing the Denominator

To rationalize a denominator, particularly when it involves a binomial expression with a square root, a systematic approach is essential. This section provides a step-by-step guide to simplify such expressions effectively. We will use the example of $\frac{17}{13-\sqrt{10}}$ to illustrate each step.

Step 1: Identify the Denominator

The first step is to clearly identify the denominator of the given fraction. In our example, the denominator is $13 - \sqrt{10}$. This is a binomial expression that includes a radical term, specifically a square root. Recognizing the structure of the denominator is crucial for determining the correct approach to rationalize it.

Step 2: Find the Conjugate

The next step is to find the conjugate of the denominator. The conjugate of a binomial expression $a - \sqrt{b}$ is $a + \sqrt{b}$, and vice versa. The conjugate is formed by changing the sign between the rational term and the square root term. For our example, the conjugate of $13 - \sqrt{10}$ is $13 + \sqrt{10}$. The use of the conjugate is pivotal because when multiplied by the original denominator, it eliminates the square root due to the difference of squares identity: $(a - b)(a + b) = a^2 - b^2$.

Step 3: Multiply Numerator and Denominator by the Conjugate

To rationalize the denominator, multiply both the numerator and the denominator of the fraction by the conjugate. This step is crucial because it maintains the value of the fraction while transforming its form. For the expression $\frac{17}{13-\sqrt{10}}$, we multiply both the numerator and the denominator by $13 + \sqrt{10}$:$\frac{17}{13-\sqrt{10}} \times \frac{13+\sqrt{10}}{13+\sqrt{10}}$

Step 4: Simplify the Expression

After multiplying by the conjugate, simplify the expression. First, multiply out the numerator:$17 \times (13 + \sqrt{10}) = 17 \times 13 + 17 \sqrt{10} = 221 + 17\sqrt{10}$

Next, multiply out the denominator using the difference of squares formula, $(a - b)(a + b) = a^2 - b^2$:

$$(13 - \sqrt{10})(13 + \sqrt{10}) = 13^2 - (\sqrt{10})^2 = 169 - 10 = 159$$

Thus, the fraction becomes:$\frac{221 + 17\sqrt{10}}{159}$

Step 5: Check for Further Simplification

Finally, check if the simplified fraction can be further reduced. Look for common factors between the coefficients in the numerator and the denominator. In our case, the fraction is$\frac{221 + 17\sqrt{10}}{159}$

Here, 221, 17, and 159 do not share any common factors other than 1. Therefore, the fraction is already in its simplest form. Following these steps methodically ensures accurate rationalization of denominators, leading to simplified and standardized mathematical expressions.

Applying the Steps to the Given Expression

Now, let's apply the step-by-step guide to rationalize the denominator of the specific expression given: $\frac{17}{13-\sqrt{10}}$. This example will reinforce the methodology discussed and provide a clear, practical demonstration of the process.

Step 1: Identify the Denominator

The denominator of the expression is $13 - \sqrt{10}$. This binomial expression includes a rational term (13) and an irrational term ($-\sqrt{10}$).

Step 2: Find the Conjugate

The conjugate of the denominator $13 - \sqrt{10}$ is $13 + \sqrt{10}$. To find the conjugate, we change the sign between the rational and irrational terms.

Step 3: Multiply Numerator and Denominator by the Conjugate

Multiply both the numerator and the denominator by the conjugate, $13 + \sqrt{10}$:$\frac{17}{13-\sqrt{10}} \times \frac{13+\sqrt{10}}{13+\sqrt{10}}$

This step ensures that we are multiplying the fraction by an equivalent form of 1, thus preserving its value while changing its form.

Step 4: Simplify the Expression

First, simplify the numerator:

$$17 \times (13 + \sqrt{10}) = 17 \times 13 + 17 \sqrt{10} = 221 + 17\sqrt{10}$$

Next, simplify the denominator using the difference of squares formula, $(a - b)(a + b) = a^2 - b^2$:

$$(13 - \sqrt{10})(13 + \sqrt{10}) = 13^2 - (\sqrt{10})^2 = 169 - 10 = 159$$

Now, the expression becomes:$\frac{221 + 17\sqrt{10}}{159}$

Step 5: Check for Further Simplification

Finally, check if the simplified fraction can be further reduced. We need to see if there are any common factors between 221, 17, and 159. The prime factorization of these numbers can help determine this.

• 221 = 13 \times 17
• 17 = 17
• 159 = 3 \times 53

Since there are no common factors among 221, 17, and 159, the fraction is already in its simplest form. Therefore, the rationalized and simplified expression is:$\frac{221 + 17\sqrt{10}}{159}$

This detailed application of the step-by-step guide clearly demonstrates how to rationalize the denominator of the given expression and simplify it to its final form.

Common Mistakes and How to Avoid Them

Rationalizing the denominator is a fundamental skill in algebra, but it's also an area where students often make mistakes. Understanding these common pitfalls and learning how to avoid them is crucial for mastering this technique. This section highlights frequent errors and provides strategies to ensure accuracy in the rationalization process.

Mistake 1: Forgetting to Multiply Both Numerator and Denominator

One of the most common mistakes is multiplying only the denominator by the conjugate and forgetting to multiply the numerator as well. This changes the value of the expression, leading to an incorrect answer. To avoid this, always remember that rationalizing the denominator involves multiplying the entire fraction by a form of 1, which means multiplying both the numerator and the denominator by the same expression (the conjugate). For example, if you have $\frac{1}{\sqrt{2}}$, you must multiply both the numerator and the denominator by $\sqrt{2}$, resulting in $\frac{\sqrt{2}}{2}$.

Mistake 2: Incorrectly Identifying the Conjugate

Another common error is misidentifying the conjugate of the denominator. The conjugate is formed by changing the sign between the terms in the binomial. For instance, the conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$, and vice versa. A frequent mistake is to change the sign of both terms or to leave the sign unchanged. To avoid this, carefully identify the binomial and ensure you are only changing the sign between the rational and irrational terms. For example, the conjugate of $3 - \sqrt{5}$ is $3 + \sqrt{5}$, not $-3 - \sqrt{5}$ or $3 - \sqrt{5}$.

Mistake 3: Incorrectly Applying the Difference of Squares

When the denominator is a binomial involving a square root, the difference of squares identity, $(a - b)(a + b) = a^2 - b^2$, is used to eliminate the square root. Errors often arise when applying this identity incorrectly. For example, students may forget to square both terms or may mishandle the subtraction. Consider the expression $(2 + \sqrt{3})(2 - \sqrt{3})$. The correct application of the difference of squares is $2^2 - (\sqrt{3})^2 = 4 - 3 = 1$. A common mistake is to calculate $2^2 + (\sqrt{3})^2$ or to make errors in squaring the terms.

Mistake 4: Failing to Simplify the Final Expression

After rationalizing the denominator, it's essential to simplify the resulting expression as much as possible. This often involves reducing fractions or simplifying square roots. A common mistake is to stop at the rationalized form without checking for further simplification. For example, after rationalizing, you might obtain $\frac{4 + 2\sqrt{2}}{6}$. This can be simplified by dividing each term in the numerator and the denominator by their greatest common divisor, which is 2, resulting in $\frac{2 + \sqrt{2}}{3}$.

Mistake 5: Distributive Property Errors

When multiplying the numerator by the conjugate, the distributive property must be applied correctly. Errors occur when students fail to distribute the multiplication across all terms in the numerator. For instance, if you have $\frac{5}{2 - \sqrt{3}}$ and you multiply the numerator by the conjugate $2 + \sqrt{3}$, the correct result is $5(2 + \sqrt{3}) = 10 + 5\sqrt{3}$. A common mistake is to write $10 + \sqrt{3}$ or $5 \times 2 + \sqrt{3}$.

By being aware of these common mistakes and consistently applying the correct techniques, students can improve their accuracy and confidence in rationalizing denominators. Regular practice and careful attention to detail are key to mastering this essential algebraic skill.

Conclusion

In conclusion, rationalizing the denominator is a vital technique in mathematics that simplifies expressions by eliminating radicals from the denominator. This process not only adheres to standard mathematical conventions but also facilitates easier manipulation and interpretation of expressions. Throughout this article, we have explored the fundamental principles behind rationalizing denominators, focusing on expressions involving square roots. We have provided a detailed, step-by-step guide, illustrating each stage with the example of $\frac{17}{13-\sqrt{10}}$, ensuring a clear and practical understanding of the methodology. The process involves identifying the denominator, finding its conjugate, multiplying both the numerator and denominator by the conjugate, simplifying the resulting expression, and checking for further simplification. Mastering these steps is crucial for solving a wide range of mathematical problems efficiently and accurately. Moreover, we have addressed common mistakes that students often make when rationalizing denominators, such as forgetting to multiply both the numerator and denominator, misidentifying the conjugate, incorrectly applying the difference of squares, failing to simplify the final expression, and making distributive property errors. By recognizing these pitfalls and implementing the strategies discussed, students can enhance their proficiency and avoid these common errors. Rationalizing denominators is not just a standalone skill; it is integral to various mathematical concepts and applications, including simplifying complex fractions, solving equations, and performing calculus operations. A solid grasp of this technique empowers students to tackle more advanced mathematical challenges with confidence and precision. Consistent practice and a methodical approach are key to mastering rationalizing denominators and ensuring success in algebra and beyond. The ability to simplify and standardize mathematical expressions is a valuable asset, and rationalizing the denominator is a fundamental step in achieving this goal.