Solving Mathematical Expressions A Step-by-Step Guide

JU07/16/2025 08, 2025 by THE IDEN 54 views

Understanding and Solving Mathematical Expressions

Mathematics can be a fascinating journey, especially when you delve into the world of expressions and equations. In this comprehensive guide, we will dissect two intriguing mathematical problems, providing a step-by-step solution and offering deeper insights into the underlying principles. Whether you are a student looking to enhance your understanding or simply someone who enjoys the beauty of numbers, this article will equip you with the knowledge and skills to tackle similar challenges. We will begin by addressing the first problem, which involves simplifying a numerical expression with integers. Then, we will move on to the second problem, which requires us to evaluate a nested radical expression. Let’s dive in and explore the world of mathematical expressions together.

H2: Problem 1 Solving Integer Expressions

The first problem we encounter is a numerical expression involving integers: -6 + (-5 + 3 - 7) - 3. This expression may seem daunting at first glance, but with a systematic approach, we can simplify it step by step. The key to solving such expressions lies in following the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). By adhering to this order, we can ensure that we arrive at the correct answer. Let's break down the expression and solve it together, highlighting the importance of each step and the underlying mathematical principles at play. This approach will not only help us solve this particular problem but also equip us with a general strategy for tackling similar expressions in the future.

H3: Step-by-Step Solution

Our initial expression is: -6 + (-5 + 3 - 7) - 3.

Parentheses First: According to PEMDAS, we must first simplify the expression within the parentheses. Inside the parentheses, we have -5 + 3 - 7. Let's break this down:
- -5 + 3 = -2
- -2 - 7 = -9 So, the expression inside the parentheses simplifies to -9.
Rewrite the Expression: Now we can rewrite the original expression with the simplified parentheses:
- -6 + (-9) - 3
Addition and Subtraction: Next, we perform addition and subtraction from left to right:
- -6 + (-9) = -15
- -15 - 3 = -18

Therefore, the simplified value of the expression is -18.

H3: Choosing the Correct Option

Looking at the options provided:

a. -18
b. 15
c. 16
d. -10

We can confidently select option a. -18 as the correct answer. This step-by-step solution demonstrates how crucial it is to follow the order of operations to arrive at the accurate result. By breaking down complex expressions into smaller, manageable parts, we can avoid errors and gain a clearer understanding of the mathematical processes involved. This approach is not only effective for solving numerical expressions but also forms the foundation for more advanced mathematical concepts.

H2: Problem 2 Evaluating Nested Radicals

The second problem presents us with a nested radical expression: ${ \sqrt[4]{\sqrt{16} - (\sqrt{2})^4} }$ . This expression involves roots within roots, which might seem intimidating at first. However, by carefully applying the properties of radicals and exponents, we can simplify this expression and find its value. The key here is to work from the innermost part of the expression outwards, simplifying each layer step by step. This systematic approach will not only lead us to the correct answer but also enhance our understanding of how radicals and exponents interact. Let’s break down this problem and explore the fascinating world of nested radicals.

H3: Step-by-Step Solution

Let's evaluate the expression ${ \sqrt[4]{\sqrt{16} - (\sqrt{2})^4} }$ step by step.

Innermost Radical: We start by simplifying the innermost radical, which is ${ \sqrt{16} }$ $16$ . The square root of 16 is 4.
- ${ \sqrt{16} = 4 }$
Exponent: Next, we evaluate the exponential term, which is ${ (\sqrt{2})^4 }$ $(2)^{4}$ . To simplify this, we can think of it as ${ (\sqrt{2})^2 * (\sqrt{2})^2 }$ $(2)^{2} * (2)^{2}$ . We know that ${ (\sqrt{2})^2 = 2 }$ $(2)^{2} = 2$ , so:
- ${ (\sqrt{2})^4 = 2 * 2 = 4 }$
Rewrite the Expression: Now we can rewrite the original expression with these simplified values:
- ${ \sqrt[4]{4 - 4} }$
Subtraction: Perform the subtraction inside the fourth root:
- ${ \sqrt[4]{0} }$
Fourth Root: Finally, we evaluate the fourth root of 0. The fourth root of 0 is 0, since 0 raised to any positive power is 0.
- ${ \sqrt[4]{0} = 0 }$

Therefore, the value of the expression is 0.

H3: Identifying the Correct Answer (Revised)

Upon reviewing the provided options:

a. -4
b. 8
c. -2
d. -10

It appears there's an error in the provided options, as our calculated answer is 0, which isn't listed. The correct answer based on our calculations is 0. It is crucial to double-check both the problem and the options to ensure accuracy. In this case, the options do not reflect the correct solution. However, the step-by-step process we followed demonstrates the correct method for evaluating nested radical expressions. This highlights the importance of not only understanding the mathematical concepts but also being meticulous in the calculations to arrive at the correct answer, even if the provided options are incorrect.

H2: Conclusion Mastering Mathematical Expressions

In this article, we tackled two distinct mathematical problems, demonstrating the importance of systematic problem-solving in mathematics. The first problem involved simplifying an integer expression, where we emphasized the crucial role of the order of operations (PEMDAS). By breaking down the expression step by step, we arrived at the correct answer of -18. This exercise highlighted the significance of following a structured approach to avoid errors and ensure accuracy. The second problem presented a nested radical expression, which required us to apply the properties of radicals and exponents. Through a careful, step-by-step simplification process, we found the value of the expression to be 0. This problem not only reinforced our understanding of radicals and exponents but also illustrated the importance of meticulous calculations. While there was a discrepancy in the provided options, the core method of solving the expression remains valid.

By working through these problems, we have not only found the solutions but also gained valuable insights into the underlying mathematical principles. These skills and understanding are essential for tackling more complex mathematical challenges in the future. Remember, mathematics is not just about finding the right answer; it’s about the journey of understanding the logic and processes involved. Keep practicing, keep exploring, and you'll continue to master the fascinating world of mathematical expressions. Whether you're a student, a professional, or simply a curious individual, the skills you develop in mathematics will serve you well in all aspects of life.