If $x = 3 + \sqrt{8}$ And $y = 3 - \sqrt{8}$, Then Find $\frac{1}{x^2} + \frac{1}{y^2}$.

Introduction

In this article, we will delve into the world of algebra and explore a complex expression involving square roots and fractions. We will use the given values of $x$ and $y$ to find the value of $\frac{1}{x^2} + \frac{1}{y^2}$. This problem requires a deep understanding of algebraic manipulations and the application of mathematical concepts.

Given Values

We are given two expressions:

$$x = 3 + \sqrt{8}$$

$$y = 3 - \sqrt{8}$$

Our goal is to find the value of $\frac{1}{x^2} + \frac{1}{y^2}$ using these given values.

Simplifying the Expressions

To simplify the expressions, we can start by finding the value of $x^2$ and $y^2$.

$$x^2 = (3 + \sqrt{8})^2$$

$$y^2 = (3 - \sqrt{8})^2$$

Using the formula $(a+b)^2 = a^2 + 2ab + b^2$, we can expand the expressions:

$$x^2 = 3^2 + 2(3)(\sqrt{8}) + (\sqrt{8})^2$$

$$y^2 = 3^2 - 2(3)(\sqrt{8}) + (\sqrt{8})^2$$

Simplifying further, we get:

$$x^2 = 9 + 12\sqrt{2} + 8$$

$$y^2 = 9 - 12\sqrt{2} + 8$$

Combining like terms, we get:

$$x^2 = 17 + 12\sqrt{2}$$

$$y^2 = 17 - 12\sqrt{2}$$

Finding the Reciprocals

Now that we have the values of $x^2$ and $y^2$, we can find the reciprocals:

$$\frac{1}{x^2} = \frac{1}{17 + 12\sqrt{2}}$$

$$\frac{1}{y^2} = \frac{1}{17 - 12\sqrt{2}}$$

To add these fractions, we need to find a common denominator. The common denominator is the product of the two denominators:

$$(17 + 12\sqrt{2})(17 - 12\sqrt{2})$$

Using the difference of squares formula, we can simplify the expression:

$$(17 + 12\sqrt{2})(17 - 12\sqrt{2}) = 17^2 - (12\sqrt{2})^2$$

Simplifying further, we get:

$$(17 + 12\sqrt{2})(17 - 12\sqrt{2}) = 289 - 288$$

Combining like terms, we get:

$$(17 + 12\sqrt{2})(17 - 12\sqrt{2}) = 1$$

Now that we have the common denominator, we can add the fractions:

$$\frac{1}{x^2} + \frac{1}{y^2} = \frac{1}{17 + 12\sqrt{2}} + \frac{1}{17 - 12\sqrt{2}}$$

Using the formula $\frac{1}{a} + \frac{1}{b} = \frac{a+b}{ab}$, we can simplify the expression:

$$\frac{1}{x^2} + \frac{1}{y^2} = \frac{(17 + 12\sqrt{2}) + (17 - 12\sqrt{2})}{(17 + 12\sqrt{2})(17 - 12\sqrt{2})}$$

Simplifying further, we get:

$$\frac{1}{x^2} + \frac{1}{y^2} = \frac{34}{1}$$

Combining like terms, we get:

$$\frac{1}{x^2} + \frac{1}{y^2} = 34$$

Conclusion

In this article, we used the given values of $x$ and $y$ to find the value of $\frac{1}{x^2} + \frac{1}{y^2}$. We simplified the expressions, found the reciprocals, and added the fractions to get the final answer. This problem required a deep understanding of algebraic manipulations and the application of mathematical concepts.

Final Answer

$$

Introduction

In our previous article, we explored a complex algebraic expression involving square roots and fractions. We used the given values of $x$ and $y$ to find the value of $\frac{1}{x^2} + \frac{1}{y^2}$. In this article, we will answer some frequently asked questions related to this problem.

Q: What is the main concept behind solving this problem?

A: The main concept behind solving this problem is the application of algebraic manipulations, including simplifying expressions, finding reciprocals, and adding fractions.

Q: Why do we need to find the reciprocals of $x^2$ and $y^2$?

A: We need to find the reciprocals of $x^2$ and $y^2$ because we are looking for the value of $\frac{1}{x^2} + \frac{1}{y^2}$. The reciprocals are the inverse of the original expressions, and we need to add them to get the final answer.

Q: How do we simplify the expressions $x^2$ and $y^2$?

A: We simplify the expressions $x^2$ and $y^2$ by using the formula $(a+b)^2 = a^2 + 2ab + b^2$. This formula allows us to expand the expressions and combine like terms.

Q: What is the common denominator for adding the fractions?

A: The common denominator for adding the fractions is the product of the two denominators, which is $(17 + 12\sqrt{2})(17 - 12\sqrt{2})$. This is a key step in adding the fractions.

Q: How do we add the fractions?

A: We add the fractions by using the formula $\frac{1}{a} + \frac{1}{b} = \frac{a+b}{ab}$. This formula allows us to simplify the expression and get the final answer.

Q: What is the final answer?

A: The final answer is $\boxed{34}$.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

• Not simplifying the expressions $x^2$ and $y^2$ correctly
• Not finding the reciprocals of $x^2$ and $y^2$ correctly
• Not using the correct common denominator when adding the fractions
• Not simplifying the expression correctly when adding the fractions

Q: How can I practice solving this type of problem?

A: You can practice solving this type of problem by working through similar examples and exercises. You can also try to come up with your own examples and exercises to practice solving.

Conclusion

In this article, we answered some frequently asked questions related to solving a complex algebraic expression involving square roots and fractions. We covered topics such as simplifying expressions, finding reciprocals, and adding fractions. We also discussed common mistakes to avoid and how to practice solving this type of problem.

Final Answer

The final answer is $\boxed{34}$.