Simplifying 7/3 + 5/2√3 - 4 + 1/4√3 - 1/2 - 3/2√3 A Step-by-Step Guide

In the realm of mathematics, simplifying expressions is a fundamental skill. This article delves into the process of simplifying an expression involving fractions and radicals, providing a comprehensive guide for students and enthusiasts alike. We will dissect the expression, identify key steps, and elucidate the underlying principles. Let's embark on this mathematical journey!

Understanding the Expression

Our focus is on simplifying the expression: 7/3 + 5/2√3 - 4 + 1/4√3 - 1/2 - 3/2√3. This expression combines rational numbers (fractions) and irrational numbers (radicals), specifically those involving the square root of 3. To simplify it effectively, we need to understand the rules for adding and subtracting fractions and how to combine like terms involving radicals.

The first step is to identify the different types of terms present. We have fractions without radicals (7/3, -4, -1/2), and terms involving the square root of 3 (5/2√3, 1/4√3, -3/2√3). Our strategy will be to combine the like terms separately. This means adding the fractions together and then adding the terms with √3 together.

To add or subtract fractions, they must have a common denominator. The fractions we have are 7/3, -4 (which can be written as -4/1), and -1/2. The least common denominator (LCD) for 3, 1, and 2 is 6. We will convert each fraction to an equivalent fraction with a denominator of 6. This involves multiplying the numerator and denominator of each fraction by the same number to achieve the desired denominator.

For the radical terms, we can treat √3 as a variable, similar to how we would combine 'x' terms in an algebraic expression. We simply add or subtract the coefficients of the √3 terms. The coefficients are 5/2, 1/4, and -3/2. Again, we need to find a common denominator to add these fractions. The LCD for 2 and 4 is 4.

By systematically addressing each part of the expression, we can simplify it step by step. This approach not only makes the problem more manageable but also reinforces the fundamental principles of mathematical operations.

Combining Rational Numbers

The initial stage in simplifying the expression involves combining the rational numbers. We need to add and subtract the fractions: 7/3, -4, and -1/2. As previously mentioned, the least common denominator (LCD) for these fractions is 6. Therefore, we must convert each fraction to an equivalent fraction with a denominator of 6.

Let's start with 7/3. To convert this fraction to have a denominator of 6, we multiply both the numerator and the denominator by 2: (7 * 2) / (3 * 2) = 14/6.

Next, we consider -4. We can rewrite this as -4/1. To convert this to a fraction with a denominator of 6, we multiply both the numerator and the denominator by 6: (-4 * 6) / (1 * 6) = -24/6.

Finally, we have -1/2. To convert this to a fraction with a denominator of 6, we multiply both the numerator and the denominator by 3: (-1 * 3) / (2 * 3) = -3/6.

Now that we have all the fractions with a common denominator, we can add and subtract them: 14/6 - 24/6 - 3/6. This is equivalent to (14 - 24 - 3) / 6. Performing the arithmetic in the numerator, we get 14 - 24 - 3 = -13. Therefore, the result of combining the rational numbers is -13/6.

This process of finding a common denominator and then adding or subtracting the numerators is a cornerstone of fraction arithmetic. It's crucial to understand this step thoroughly, as it applies to a wide range of mathematical problems. By converting each fraction to an equivalent form with the same denominator, we ensure that we are adding or subtracting comparable quantities. The result, -13/6, represents the simplified form of the rational part of the original expression.

Simplifying Radical Terms

After addressing the rational numbers, the next crucial step is simplifying the radical terms. In our expression, these terms are 5/2√3, 1/4√3, and -3/2√3. To combine these terms effectively, we treat √3 as a common factor, similar to a variable in an algebraic expression. This allows us to focus on the coefficients of √3, which are 5/2, 1/4, and -3/2.

To add or subtract these coefficients, we need to find a common denominator. The least common denominator (LCD) for the denominators 2 and 4 is 4. Therefore, we will convert each fraction to an equivalent fraction with a denominator of 4.

Let's start with 5/2. To convert this fraction to have a denominator of 4, we multiply both the numerator and the denominator by 2: (5 * 2) / (2 * 2) = 10/4. So, 5/2√3 becomes 10/4√3.

Next, we have 1/4√3, which already has a denominator of 4, so it remains as 1/4√3.

Finally, we consider -3/2. To convert this to a fraction with a denominator of 4, we multiply both the numerator and the denominator by 2: (-3 * 2) / (2 * 2) = -6/4. Thus, -3/2√3 becomes -6/4√3.

Now that all the coefficients have a common denominator, we can combine them: 10/4√3 + 1/4√3 - 6/4√3. This is equivalent to (10/4 + 1/4 - 6/4)√3. Performing the arithmetic within the parentheses, we have (10 + 1 - 6) / 4 = 5/4. Therefore, the simplified form of the radical terms is 5/4√3.

This process highlights the importance of treating radicals as variables when combining like terms. By focusing on the coefficients and using the rules of fraction arithmetic, we can effectively simplify expressions involving radicals. The result, 5/4√3, represents the simplified form of the radical part of the original expression.

Final Simplified Expression

Having simplified both the rational and radical parts of the expression, we are now ready to present the final simplified expression. We found that the rational part simplifies to -13/6, and the radical part simplifies to 5/4√3. To combine these, we simply add them together.

Therefore, the final simplified expression is -13/6 + 5/4√3. This expression represents the most concise form of the original expression, combining the rational and irrational components into a single, simplified result.

It's important to note that we cannot combine -13/6 and 5/4√3 any further because they are not like terms. The first term is a rational number, while the second term is an irrational number multiplied by a radical. They are fundamentally different types of numbers and cannot be added or subtracted directly.

The final simplified expression, -13/6 + 5/4√3, demonstrates the power of mathematical simplification. By systematically breaking down the expression into its components, applying the rules of fraction arithmetic, and combining like terms, we have arrived at a more manageable and understandable form. This process not only simplifies the expression but also enhances our understanding of the underlying mathematical principles.

In conclusion, simplifying expressions involving fractions and radicals requires a methodical approach. By identifying like terms, finding common denominators, and applying the rules of arithmetic, we can effectively reduce complex expressions to their simplest forms. This skill is essential for success in mathematics and provides a foundation for more advanced concepts.