Evaluating Mathematical Expressions A Step By Step Guide

This article provides a detailed walkthrough on evaluating various mathematical expressions, focusing on simplifying expressions involving square roots, fractions, and basic arithmetic operations. We will dissect each expression step by step, ensuring clarity and understanding of the underlying mathematical principles. This guide aims to enhance your problem-solving skills and build a solid foundation in mathematical computations. Dive in and let’s explore the fascinating world of mathematical expressions!

1. Evaluating the Expression: (6 + √9) / (2(3))

The initial expression we'll tackle is (6 + √9) / (2(3)). This expression combines several fundamental arithmetic operations, including addition, square root calculation, multiplication, and division. To accurately evaluate this, we need to follow the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Understanding and applying PEMDAS is crucial in simplifying any mathematical expression effectively. Let's begin breaking down this expression step-by-step.

First, we need to address the square root within the expression. The square root of 9 (√9) is 3, since 3 multiplied by itself equals 9. Therefore, we can replace √9 with 3 in our expression. This simplification is a critical step, as it allows us to move forward with the other operations more easily. Simplifying square roots is a common technique in mathematical problem-solving, and mastering it is essential for handling more complex problems.

Now our expression looks like this: (6 + 3) / (2(3)). The next operation according to PEMDAS is to handle the parentheses. Inside the first set of parentheses, we have 6 + 3, which equals 9. Inside the second set of parentheses, we have 2(3), which signifies multiplication. Multiplying 2 by 3 gives us 6. This step-by-step approach ensures that we are breaking down the expression into manageable parts. Handling parentheses correctly is vital to avoid errors in the final result.

After simplifying the parentheses, our expression becomes 9 / 6. This is a simple division problem. Dividing 9 by 6 gives us 1.5, which can also be expressed as the fraction 3/2. This final step demonstrates how we reduce the expression to its simplest form. Converting decimals to fractions and vice versa is a useful skill in mathematical simplifications.

In summary, evaluating (6 + √9) / (2(3)) involves understanding the order of operations, simplifying square roots, performing addition and multiplication within parentheses, and finally, dividing. The correct sequence leads us to the simplified answer of 1.5 or 3/2. This process underscores the importance of meticulous step-by-step calculation in mathematics.

2. Evaluating the Expression: (6 - √9) / (2(3))

Next, let's delve into evaluating the expression (6 - √9) / (2(3)). Similar to the first expression, this one involves square roots, parentheses, subtraction, multiplication, and division. The key to accurately evaluating this expression lies in adhering to the correct order of operations, following the PEMDAS rule. This ensures that we simplify the expression in a logical and mathematically sound manner. Mastering the order of operations is fundamental for solving any mathematical problem effectively.

As before, we start with the square root. The square root of 9 (√9) is 3. Substituting this value into our expression, we get (6 - 3) / (2(3)). This initial step of simplifying the square root is crucial as it allows us to proceed with the other operations more smoothly. Simplifying radicals is a common practice in mathematics and is an essential skill to develop.

Now, we focus on the parentheses. In the numerator, we have (6 - 3), which equals 3. In the denominator, we have (2(3)), indicating multiplication, which gives us 6. Our expression now simplifies to 3 / 6. This methodical approach to simplifying each part of the expression is vital to avoid errors. Handling parentheses and their operations accurately is a key step in mathematical simplifications.

Finally, we perform the division. Dividing 3 by 6 gives us 0.5, which can also be expressed as the fraction 1/2. This final simplification provides the answer in its simplest form. Understanding how to convert between decimals and fractions is a valuable skill in mathematics. Converting fractions to decimals and vice versa often helps in understanding the magnitude of the result.

Thus, the step-by-step evaluation of (6 - √9) / (2(3)) involves simplifying the square root, performing subtraction and multiplication within parentheses, and then dividing. Following this sequence, we arrive at the simplified result of 0.5 or 1/2. This demonstrates the importance of following a structured approach to mathematical problem-solving.

3. Evaluating the Expression: (-6 + √18) / (2(2))

Now, let’s tackle the expression (-6 + √18) / (2(2)). This expression introduces a slightly more complex square root, √18, which is not a perfect square. We'll need to simplify it before proceeding with the rest of the calculation. Additionally, this expression includes negative numbers, so careful attention to sign conventions is important. Dealing with negative numbers and non-perfect square roots is a crucial aspect of mathematical proficiency.

First, we need to simplify √18. We can express 18 as 9 * 2, where 9 is a perfect square. Therefore, √18 can be written as √(9 * 2), which simplifies to √9 * √2, or 3√2. Substituting this back into the expression, we get (-6 + 3√2) / (2(2)). Simplifying radicals often involves finding perfect square factors. Simplifying square roots by identifying perfect square factors is a key technique in algebra.

Next, we look at the denominator. The expression 2(2) is a simple multiplication, which equals 4. So, our expression now looks like (-6 + 3√2) / 4. There are no more simplifications we can do directly, but it's important to assess if we can factor out any common terms in the numerator to simplify further. Looking for common factors is a useful step in simplifying algebraic expressions.

In this case, we can notice that -6 and 3 have a common factor of 3. We could factor out a 3 from the numerator, giving us 3(-2 + √2) / 4. However, this doesn't lead to any further simplification as there are no common factors between the numerator and the denominator. This step highlights the importance of always looking for potential simplifications. Factoring can often reveal hidden simplifications in mathematical expressions.

Therefore, the simplified form of the expression (-6 + √18) / (2(2)) is (-6 + 3√2) / 4. While we cannot reduce this expression to a single numerical value without approximating √2, this form is the most simplified representation. This example underscores the importance of understanding how to simplify square roots and recognize when an expression is in its simplest form. Recognizing simplified forms is a key aspect of mathematical problem-solving.

4. Evaluating the Expression: (-9 - √24) / (2(2))

Let's move on to the expression (-9 - √24) / (2(2)). This expression, like the previous one, includes a square root that is not a perfect square (√24) and also involves negative numbers. The approach to evaluating this expression will be similar: we need to simplify the square root first and then perform the other operations, paying close attention to the signs. Handling non-perfect square roots and negative signs requires a methodical and careful approach.

The first step is to simplify √24. We can express 24 as 4 * 6, where 4 is a perfect square. So, √24 can be written as √(4 * 6), which simplifies to √4 * √6, or 2√6. Substituting this into our expression, we get (-9 - 2√6) / (2(2)). This simplification is a crucial step in making the expression more manageable. Simplifying radicals by finding perfect square factors is a fundamental technique in algebra.

Next, we address the denominator. The expression 2(2) is a straightforward multiplication, which equals 4. Our expression now looks like (-9 - 2√6) / 4. At this point, we need to assess if there are any common factors that we can factor out from the numerator and the denominator to further simplify the expression. Checking for common factors is always a good practice in simplification.

Looking at the numerator, -9 and -2 do not have any common factors other than 1. Therefore, we cannot simplify the expression further by factoring. This means the expression is in its simplest form. This step demonstrates that not all expressions can be simplified to a single number or a simpler fraction. Understanding the limits of simplification is an important aspect of mathematical problem-solving.

Thus, the simplified form of the expression (-9 - √24) / (2(2)) is (-9 - 2√6) / 4. We cannot simplify this expression further without approximating the value of √6. This example highlights the importance of being able to simplify square roots and recognize when an expression is in its most reduced form. Recognizing simplified forms is essential for presenting mathematical solutions accurately.

5. Evaluating the Expression: (-8 + √(64 - 28)) / (2(-3))

Let's evaluate the expression (-8 + √(64 - 28)) / (2(-3)). This expression involves a square root of a difference, which means we need to perform the subtraction inside the square root first. It also includes negative numbers and multiplication in the denominator. Dealing with square roots containing subtractions and negative numbers requires a careful application of the order of operations.

The first step is to simplify the expression inside the square root. We have 64 - 28, which equals 36. So, our expression becomes (-8 + √36) / (2(-3)). Simplifying the expression under the square root is crucial for making the problem more manageable. Simplifying expressions within radicals is a common first step in many algebraic problems.

Next, we need to evaluate the square root of 36. The square root of 36 (√36) is 6. Substituting this value into the expression, we get (-8 + 6) / (2(-3)). This simplification allows us to move forward with the remaining arithmetic operations. Evaluating square roots is a fundamental skill in algebra.

Now, we simplify the numerator and the denominator separately. In the numerator, we have -8 + 6, which equals -2. In the denominator, we have 2(-3), which signifies multiplication, resulting in -6. So, our expression now simplifies to -2 / -6. Handling operations in the numerator and denominator separately is a key step in simplifying fractions.

Finally, we perform the division. Dividing -2 by -6 gives us 1/3, as a negative divided by a negative is a positive. This is the simplest form of the expression. This final step demonstrates the importance of simplifying fractions to their lowest terms. Simplifying fractions is a fundamental skill in mathematics.

Therefore, the step-by-step evaluation of (-8 + √(64 - 28)) / (2(-3)) involves simplifying the expression inside the square root, taking the square root, simplifying the numerator and denominator, and then dividing. Following these steps, we arrive at the simplified result of 1/3. This example emphasizes the importance of following the order of operations and simplifying expressions step by step.

6. Evaluating the Expression: (-6 - √(36 - 20)) / (2(1))

Finally, let’s evaluate the expression (-6 - √(36 - 20)) / (2(1)). This expression, like the previous ones, involves a square root of a difference, negative numbers, and basic arithmetic operations. The key to solving this expression correctly is to follow the order of operations (PEMDAS) meticulously. The importance of PEMDAS cannot be overstated when evaluating mathematical expressions.

We begin by simplifying the expression inside the square root. We have 36 - 20, which equals 16. So, the expression becomes (-6 - √16) / (2(1)). Simplifying within the square root is an essential first step to make the expression easier to handle. Simplifying under the radical is often the first step in these types of problems.

Next, we evaluate the square root of 16. The square root of 16 (√16) is 4. Substituting this into our expression, we get (-6 - 4) / (2(1)). This step allows us to replace the square root with a simple number, making further calculations more straightforward. Calculating square roots is a fundamental skill.

Now, we simplify the numerator and the denominator separately. In the numerator, we have -6 - 4, which equals -10. In the denominator, we have 2(1), which is a simple multiplication resulting in 2. The expression now simplifies to -10 / 2. Separately simplifying numerator and denominator is a common technique in fraction simplification.

Finally, we perform the division. Dividing -10 by 2 gives us -5. This is the simplest form of the expression. This final step highlights the importance of performing division as the last step in the simplification process. Performing division is the final step in many mathematical simplifications.

Thus, evaluating (-6 - √(36 - 20)) / (2(1)) involves simplifying the expression inside the square root, calculating the square root, simplifying the numerator and denominator, and then dividing. By following these steps, we arrive at the simplified result of -5. This example reinforces the need for a step-by-step approach and careful attention to detail when working with mathematical expressions.

Conclusion

In conclusion, evaluating mathematical expressions requires a solid understanding of the order of operations, the ability to simplify square roots, and careful attention to detail. Each expression we tackled in this article demonstrated different facets of mathematical simplification, from dealing with parentheses and negative numbers to handling non-perfect square roots. By breaking down each expression step by step, we were able to arrive at the simplified result, highlighting the importance of a methodical approach in mathematics. This guide serves as a comprehensive resource for enhancing your mathematical problem-solving skills and building a strong foundation for more advanced concepts.