Wyznacz Liczbę X, Której 80% Wynosi 2^3 + 2√3 / 4√3. Podaj Wynik.

Introduction

In this comprehensive article, we will delve into a mathematical problem that involves determining a number based on a given percentage and an algebraic expression. Specifically, we aim to find the number for which 80% equals the result of the expression 2^3 + 2√3 divided by 4√3. This problem combines basic arithmetic operations with algebraic concepts and percentage calculations, making it an excellent exercise for strengthening mathematical skills. To effectively tackle this problem, we will break it down into smaller, manageable steps. This approach will not only simplify the solution process but also enhance understanding. We'll begin by evaluating the algebraic expression, then use the result to find the unknown number based on the given percentage. This method allows us to address each component systematically, ensuring accuracy and clarity in our calculations. Throughout this detailed explanation, we will emphasize each step, providing a clear and concise solution. This problem is a great example of how different mathematical principles can be integrated to solve a single question. By working through it methodically, readers can gain confidence in their ability to handle similar problems and improve their overall mathematical proficiency. The final answer will be presented clearly, along with a summary of the steps taken to reach the solution. This will provide a comprehensive understanding and serve as a useful reference for future calculations.

Step 1: Evaluate the Expression 2^3 + 2√3 Divided by 4√3

To begin, we need to evaluate the given expression: (2^3 + 2√3) / (4√3). This involves simplifying both the numerator and the denominator before performing the division. This initial simplification is crucial because it sets the stage for accurate calculation and helps to minimize potential errors. A clear understanding of each component of the expression is essential. First, we'll tackle the numerator. 2^3 means 2 multiplied by itself three times, which equals 8. Therefore, the numerator becomes 8 + 2√3. This part of the simplification combines a whole number with an irrational number, which we will handle carefully as we proceed. Next, we look at the denominator, which is 4√3. This term includes a whole number multiplied by the square root of 3. The presence of the square root in both the numerator and the denominator suggests that further simplification might be possible, particularly if we can find a common factor or manipulate the expression to eliminate the square root from the denominator. To effectively simplify the entire expression, we will explore methods such as rationalizing the denominator or factoring out common terms. Each of these techniques will help us reduce the expression to its simplest form. This step is not just about finding the numerical value; it’s about understanding how algebraic expressions can be manipulated to make them easier to work with. This skill is fundamental in mathematics and will be valuable in solving a wide range of problems. By carefully evaluating and simplifying this expression, we lay the groundwork for the subsequent steps in finding the number that corresponds to 80% of the result.

Simplify the Numerator

Let's break down the numerator further: 2^3 + 2√3. As mentioned earlier, 2^3 equals 8. So, the numerator becomes 8 + 2√3. Now, we look for common factors. We can see that both terms, 8 and 2√3, have a common factor of 2. Factoring out the 2, we get 2(4 + √3). This step is crucial because it simplifies the expression and makes it easier to handle in further calculations. The factored form of the numerator highlights the relationship between the whole number part and the irrational part, making subsequent steps more straightforward. Factoring is a key technique in algebra, allowing us to rewrite expressions in a more manageable form. It helps in simplifying complex expressions, solving equations, and identifying common terms that can be canceled out. In this case, factoring out the 2 from the numerator not only simplifies the expression but also prepares it for potential cancellation or further simplification when we consider the denominator. The ability to recognize and apply factoring techniques is essential for solving mathematical problems efficiently. By simplifying the numerator in this manner, we are setting ourselves up for a more manageable division in the next step. This meticulous approach to simplification is a hallmark of effective problem-solving in mathematics.

Simplify the Denominator

The denominator in our expression is 4√3. This term is already in a relatively simplified form, but it's important to recognize that it contains a square root. In many mathematical contexts, it's preferable to rationalize the denominator, meaning we eliminate the square root from the denominator. However, in this case, we will proceed with the simplified form of the denominator as it is, keeping in mind that we might need to rationalize it later if further simplification requires it. The key is to understand the structure of the denominator and how it interacts with the numerator. Since the numerator now contains a term with √3, we might be able to simplify the entire expression by either canceling out common factors or rationalizing the denominator. Rationalizing the denominator involves multiplying both the numerator and the denominator by the square root term in the denominator. This process eliminates the square root from the denominator but might introduce it into the numerator. The decision to rationalize the denominator depends on the overall goal of the simplification and the context of the problem. In this instance, we will proceed without rationalizing initially, but we remain prepared to do so if necessary. The understanding of when and how to rationalize denominators is a critical skill in algebra, as it often leads to simpler and more manageable expressions. By carefully considering the structure of both the numerator and the denominator, we can make informed decisions about the best approach to simplification.

Divide the Simplified Numerator by the Simplified Denominator

Now, let's divide the simplified numerator by the denominator: (2(4 + √3)) / (4√3). This step involves dividing the entire factored numerator by the denominator. We can begin by simplifying the fraction by canceling out common factors. Notice that both the numerator and the denominator have a factor of 2. Dividing both by 2, we get (4 + √3) / (2√3). This simplification makes the expression more manageable. Next, to further simplify the expression, we rationalize the denominator. To do this, we multiply both the numerator and the denominator by √3. This process will eliminate the square root from the denominator, making the expression easier to work with. When we multiply the numerator (4 + √3) by √3, we get 4√3 + 3. Multiplying the denominator 2√3 by √3 gives us 2 * 3 = 6. So, the expression becomes (4√3 + 3) / 6. This rationalized form is often preferred in mathematical expressions because it avoids having a square root in the denominator. The process of rationalizing the denominator is a standard technique in algebra, and it’s essential for simplifying expressions and performing further calculations. By dividing the simplified numerator by the denominator and rationalizing the denominator, we have transformed the original expression into a more usable form. This simplified expression will now allow us to find the number for which 80% equals this result.

Result of the Division

After dividing the simplified numerator by the denominator and rationalizing, we have the expression (4√3 + 3) / 6. This is the simplified result of the original expression (2^3 + 2√3) / (4√3). Now, we need to approximate the numerical value of this expression to proceed further. We know that the square root of 3 (√3) is approximately 1.732. Using this approximation, we can substitute √3 in our expression. So, 4√3 is approximately 4 * 1.732, which equals 6.928. Adding 3 to this, we get 6.928 + 3 = 9.928. Finally, we divide 9.928 by 6. This gives us approximately 1.6547. Therefore, the value of the expression (4√3 + 3) / 6 is approximately 1.6547. This numerical approximation is crucial because it allows us to transition from an algebraic expression to a decimal value, which is necessary for the subsequent percentage calculation. Understanding how to work with approximations is essential in mathematics, especially when dealing with irrational numbers like √3. The ability to accurately approximate values ensures that our final result will be close to the true answer. By finding the approximate value of the expression, we have laid the groundwork for the next step, which involves finding the number for which 80% equals 1.6547. This methodical progression from simplifying the expression to approximating its value is a key strategy in solving mathematical problems.

Step 2: Calculate the Number for Which 80% Equals the Result

Now that we know the value of the expression (2^3 + 2√3) / (4√3) is approximately 1.6547, we can proceed to find the number for which 80% equals this value. This involves setting up an equation and solving for the unknown number. This step combines percentage calculations with algebraic problem-solving, highlighting the interconnectedness of different mathematical concepts. To begin, we can represent the unknown number as 'x'. The problem states that 80% of x is equal to 1.6547. We can write this as an equation: 0.80x = 1.6547. This equation is a simple linear equation, which can be easily solved by isolating x. Solving for x involves dividing both sides of the equation by 0.80. This will give us the value of the unknown number. The process of setting up and solving equations is a fundamental skill in algebra. It allows us to translate word problems into mathematical statements and find solutions. In this case, the ability to convert the percentage problem into an algebraic equation is crucial for finding the answer. By carefully performing the division, we will determine the number for which 80% is equal to the simplified value of our initial expression. This step demonstrates the practical application of percentage calculations and algebraic techniques in solving real-world problems. The final result will provide a clear answer to the question, completing the solution process.

Set Up the Equation

To find the number for which 80% equals approximately 1.6547, we need to translate this problem into an algebraic equation. Let's represent the unknown number as 'x'. **The statement