1) Simplify The Expression √4-2√3. 2) Prove That The Number √11-2√28 - √11+2√28 Is An Integer. 3) Find The Value Of The Expression 1/b - 1/a, Given The Condition (√3 X - √3 Y) / (...). Please Provide Explanations For Each Step.

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In this article, we will delve into the solutions of several algebraic expressions, providing detailed explanations for each step. We will cover simplifying square roots, proving integer results, and finding the value of expressions involving fractions and square roots. This comprehensive guide aims to enhance your understanding of algebraic manipulations and problem-solving techniques.

1) Simplifying the Square Root: √4-2√3

In this section, we aim to simplify the expression √4-2√3. This type of problem often involves recognizing a pattern that allows us to rewrite the expression inside the square root as a perfect square. Our main keyword here is simplifying square roots, and we will use it to guide our explanation.

To effectively simplify this expression, our primary focus is to manipulate the expression within the square root to resemble the form (a - b)². This form expands to a² - 2ab + b², which we can then rewrite as the square of a binomial. When we identify this structure, we can eliminate the square root, leading to a simplified result. The key is to recognize that the term -2√3 corresponds to the -2ab part of the expansion.

Now, let’s analyze the expression 4 - 2√3. We want to express this in the form a² - 2ab + b². Comparing -2√3 with -2ab, we can infer that ab = √3. We need to find values for a and b that satisfy this condition and also make a² + b² equal to 4. By trying different combinations, we find that if a = √3 and b = 1, then ab = √3 and a² + b² = (√3)² + 1² = 3 + 1 = 4. This is exactly what we need.

Therefore, we can rewrite the expression inside the square root as (√3)² - 2(√3)(1) + 1², which is equivalent to (√3 - 1)². Now, we have:

√4 - 2√3 = √(√3 - 1)²

Since the square root and the square cancel each other out, we get:

√4 - 2√3 = |√3 - 1|

Since √3 is greater than 1, the expression √3 - 1 is positive, so the absolute value is simply:

√4 - 2√3 = √3 - 1

Thus, the simplified form of √4 - 2√3 is √3 - 1. This result is achieved by recognizing and utilizing the perfect square trinomial pattern within the square root, allowing for a straightforward simplification.

2) Proving the Integer Result: √11-2√28 - √11+2√28

In this section, we aim to prove that the number √11-2√28 - √11+2√28 is an integer. The core of this problem lies in simplifying each square root term and then evaluating the overall expression. Our main keyword here is proving integer results, and we will use it to structure our explanation.

To demonstrate that the expression results in an integer, we first need to simplify the individual square root terms, √11-2√28 and √11+2√28. The approach we will use is similar to the first problem: we aim to rewrite the expressions inside the square roots as perfect squares. This involves recognizing the form a² ± 2ab + b² and identifying the appropriate values for a and b.

Let's start with √11-2√28. We need to express 11 - 2√28 in the form a² - 2ab + b². The term -2√28 corresponds to -2ab, which means ab = √28. We can simplify √28 as √(4 * 7) = 2√7. Thus, ab = 2√7. Now, we need to find values for a and b such that a² + b² = 11. By inspection, we can identify that if a = √7 and b = 2, then ab = 2√7 and a² + b² = (√7)² + 2² = 7 + 4 = 11. Therefore, we can rewrite 11 - 2√28 as (√7 - 2)².

So, √11-2√28 = √(√7 - 2)² = |√7 - 2|. Since √7 is greater than 2, the absolute value simplifies to √7 - 2.

Now, let's simplify √11+2√28. We need to express 11 + 2√28 in the form a² + 2ab + b². We already know that ab = 2√7. Using the same values for a and b as before, a = √7 and b = 2, we have a² + b² = 11. Thus, we can rewrite 11 + 2√28 as (√7 + 2)².

So, √11+2√28 = √(√7 + 2)² = |√7 + 2|. Since √7 + 2 is positive, the absolute value simplifies to √7 + 2.

Now we can substitute these simplified expressions back into the original expression:

√11-2√28 - √11+2√28 = (√7 - 2) - (√7 + 2)

Distributing the negative sign, we get:

√7 - 2 - √7 - 2

The √7 terms cancel each other out, leaving:

-2 - 2 = -4

Therefore, the expression √11-2√28 - √11+2√28 simplifies to -4, which is an integer. This proves that the original expression results in an integer value, demonstrating the effectiveness of simplifying square roots and recognizing perfect square patterns.

3) Finding the Value of the Expression: 1/b - 1/a

In this section, we will determine the value of the expression 1/b - 1/a, given the condition (√3 x - √3 y) / (...) = something . This problem involves algebraic manipulation and potentially some simplification using the given condition. Our main keyword here is finding the value of expressions, and we will use it to guide our approach.

To find the value of 1/b - 1/a, we first need to simplify this expression into a more manageable form. The common denominator for the two fractions is ab, so we can rewrite the expression as:

1/b - 1/a = (a - b) / ab

Now, we need to utilize the given condition to find the values or relationship between a and b. The condition provided is (√3 x - √3 y) / (...) = something. However, the condition is incomplete, missing the denominator and the result. Without the complete condition, we cannot directly find the values of a and b. We will assume a possible scenario where the condition might help us relate a and b and demonstrate how to proceed if the full condition were available.

Let's assume, for the sake of illustration, that the complete condition is:

(√3 x - √3 y) / (x - y) = √3

And that we have additional information which leads us to identify a = x and b = y. With this assumption, we can rewrite the expression (a - b) / ab as (x - y) / xy. Now, we need to find a relationship between x and y that we can use to simplify this further.

From the assumed condition, we have:

(√3 x - √3 y) / (x - y) = √3

We can factor out √3 from the numerator:

√3 (x - y) / (x - y) = √3

As long as x ≠ y, we can cancel the (x - y) terms:

√3 = √3

This does not directly give us a specific relationship between x and y that we can use to find a numerical value for (x - y) / xy. However, it confirms that our simplification is consistent.

Let's consider a different hypothetical scenario where we have additional information, such as a specific relationship between x and y. For example, suppose we know that x = 2y. Then we can substitute this into our expression:

(x - y) / xy = (2y - y) / (2y * y) = y / 2y² = 1 / 2y

If we had a specific value for y, we could find a numerical answer. For example, if y = 1, then the expression would evaluate to 1/2.

In the absence of a complete condition and additional information, we cannot find a specific numerical value for 1/b - 1/a. The process involves simplifying the expression, utilizing the given condition to find a relationship between the variables, and then substituting that relationship back into the simplified expression. This illustrates the general approach to solving such problems when complete information is available. The key is to manipulate the algebraic expressions and use the given conditions effectively to find the required value.

In conclusion, we have tackled three different types of algebraic problems: simplifying square roots, proving integer results, and finding the value of expressions. Each problem required a unique approach, utilizing algebraic manipulation techniques such as recognizing perfect square trinomials, simplifying expressions, and using given conditions to find relationships between variables. By mastering these techniques, you can effectively solve a wide range of algebraic problems. Remember to break down complex problems into smaller steps, simplify expressions where possible, and utilize all available information to reach the solution. This comprehensive approach will enhance your problem-solving skills in algebra and beyond.