Simplifying Expressions Step-by-Step Guide For $(\sqrt[3]{27}-14 \div 2)(6-8)^2)

Aug 1, 2025 by ITMIN 81 views

$Simplifying the Expression: A Step-by-Step Guide to $(\sqrt[3]{27}-14 \div 2)(6-8)^2$$

Simplifying expressions is a fundamental skill in mathematics, and it involves reducing a mathematical expression to its simplest form. This often entails performing operations, combining like terms, and applying various mathematical rules and properties. In this article, we will embark on a detailed journey to simplify the expression $(\sqrt[3]{27}-14 \div 2)(6-8)^2$ , meticulously outlining each step along the way. This comprehensive guide will not only help you understand the simplification process but also enhance your problem-solving abilities in mathematics. Understanding the step-by-step simplification process is crucial for mastering mathematical concepts and tackling more complex problems. In this detailed guide, we will break down each step involved in simplifying the expression, providing clear explanations and insights to enhance your understanding. By following along, you'll gain valuable skills in applying mathematical rules, performing operations, and reducing expressions to their simplest forms. This knowledge will not only benefit you in academic settings but also in various real-world scenarios where mathematical thinking is essential. From basic arithmetic to advanced algebra and calculus, the ability to simplify expressions is a cornerstone of mathematical proficiency. So, let's dive into the world of mathematical simplification and unravel the intricacies of this fundamental skill.

Order of Operations (PEMDAS/BODMAS)

Before we commence the simplification, it's crucial to understand the order of operations, often remembered by the acronyms PEMDAS or BODMAS. This order dictates the sequence in which we perform mathematical operations:

Parentheses / Brackets
Exponents / Orders
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)

The PEMDAS/BODMAS principle is the foundation of simplifying mathematical expressions. This mnemonic acronym serves as a guiding light, ensuring that we approach each problem with a systematic and logical order of operations. By adhering to this principle, we can prevent errors and arrive at the correct solution. Parentheses or brackets, the first step in the order, act as containers, prioritizing the operations within them. Exponents, the second step, dictate the power to which a number is raised, significantly influencing the expression's value. Multiplication and division, the third step, hold equal precedence and are executed from left to right, ensuring fairness in their application. Finally, addition and subtraction, the last step, also share equal importance and are performed from left to right, completing the simplification process. Mastery of PEMDAS/BODMAS is not just a mathematical skill; it's a cognitive tool that enhances our ability to approach complex problems methodically and effectively. Understanding this order is paramount to simplifying any mathematical expression correctly. Neglecting this order can lead to incorrect results, underscoring the importance of adhering to this fundamental principle. In our journey to simplify the given expression, we will meticulously follow the PEMDAS/BODMAS guidelines, ensuring that each operation is performed in the correct sequence. This systematic approach will not only lead us to the accurate answer but also reinforce our understanding of mathematical principles. Remember, PEMDAS/BODMAS is not just a set of rules; it's a pathway to mathematical clarity and precision.

Step 1: Simplify within Parentheses

Our initial focus is on the parentheses within the expression: $(\sqrt[3]{27}-14 \div 2)(6-8)^2$ . Let's break it down.

Parenthesis 1: $(\sqrt[3]{27}-14 \div 2)$

Here, we encounter both a cube root and division. Following PEMDAS/BODMAS, we address the cube root first:

$\sqrt[3]{27} = 3$ (since $3 imes 3 imes 3 = 27$ )

Next, we perform the division:

$14 \div 2 = 7$

Now, we substitute these values back into the parenthesis:

$3 - 7 = -4$

Parenthesis 2: $(6-8)$

This is a straightforward subtraction:

$6 - 8 = -2$

The first step in simplifying complex mathematical expressions involves carefully examining and simplifying the contents within parentheses. Parentheses act as containers, encapsulating operations that must be performed before any other calculations outside of them. This principle is rooted in the order of operations, often remembered by the acronyms PEMDAS or BODMAS, which prioritizes calculations within parentheses or brackets. By addressing the operations within parentheses first, we ensure that the expression is simplified in a logical and consistent manner, preventing potential errors and ensuring accuracy. In our journey to simplify the expression $(\sqrt[3]{27}-14 \div 2)(6-8)^2$ , we encounter two sets of parentheses, each containing its own set of operations. The first set, $(\sqrt[3]{27}-14 \div 2)$ , presents us with a cube root and a division, while the second set, $(6-8)$ , involves a simple subtraction. By meticulously simplifying each set of parentheses, we lay the foundation for further simplification and ultimately arrive at the correct answer. The process of simplifying within parentheses is not just a mathematical exercise; it's a demonstration of our ability to break down complex problems into smaller, more manageable steps. This skill is invaluable not only in mathematics but also in various other aspects of life where problem-solving is essential. So, let's embrace the power of parentheses and unlock the secrets to simplifying mathematical expressions.

Step 2: Simplify the Exponent

Our expression now looks like this: $(-4)(-2)^2$ . The next operation according to PEMDAS/BODMAS is dealing with the exponent:

$(-2)^2 = (-2) imes (-2) = 4$

Exponents, the mathematical shorthand for repeated multiplication, play a crucial role in simplifying expressions. They dictate the power to which a number, known as the base, is raised. Understanding and simplifying exponents is paramount in adhering to the order of operations and arriving at the correct solution. In our expression, $(-4)(-2)^2$ , the exponent is applied to the term $(-2)$ , indicating that we need to multiply $-2$ by itself. This operation, $(-2)^2$ , is equivalent to $(-2) imes (-2)$ , which yields a positive result of $4$ . By simplifying the exponent, we transform the expression into a more manageable form, paving the way for the final multiplication step. The process of simplifying exponents is not merely a mechanical calculation; it's a demonstration of our understanding of mathematical notation and the rules that govern it. Exponents are not just numbers; they represent a fundamental mathematical concept with wide-ranging applications in various fields, from science and engineering to finance and economics. By mastering the art of simplifying exponents, we equip ourselves with a powerful tool for tackling complex problems and unlocking the secrets of the mathematical universe. The ability to simplify exponents is not just a mathematical skill; it's a gateway to a deeper understanding of mathematical principles and their applications in the real world. So, let's embrace the power of exponents and continue our journey towards mathematical mastery.

Step 3: Perform Multiplication

We are now left with: $(-4)(4)$ . This is a simple multiplication:

$(-4)(4) = -16$

Multiplication, a fundamental arithmetic operation, plays a pivotal role in simplifying mathematical expressions. It involves combining two or more numbers, known as factors, to produce their product. In the context of simplifying expressions, multiplication often serves as the final step, bringing together the results of previous operations to arrive at the ultimate answer. In our expression, $(-4)(4)$ , we are tasked with multiplying two integers: $-4$ and $4$ . This operation is straightforward, requiring us to multiply the absolute values of the numbers and then apply the appropriate sign. The product of $4$ and $4$ is $16$ , and since we are multiplying a negative number by a positive number, the result is negative. Therefore, $(-4)(4)$ equals $-16$ . The act of multiplication is not merely a mechanical process; it's a demonstration of our understanding of number relationships and the rules that govern them. Multiplication is not just a mathematical operation; it's a fundamental building block of mathematics, with applications spanning various fields, from basic arithmetic to advanced calculus. By mastering the art of multiplication, we equip ourselves with a powerful tool for simplifying expressions and solving a wide range of mathematical problems. The ability to perform multiplication accurately and efficiently is not just a mathematical skill; it's a gateway to a deeper understanding of mathematical principles and their applications in the real world. So, let's embrace the power of multiplication and continue our journey towards mathematical mastery. Multiplication is the final step in this simplification, combining the results of the previous operations to arrive at the ultimate answer. This step showcases the culmination of our efforts, bringing together the simplified terms to reveal the final solution.

Final Answer

Therefore, the simplified form of the expression $(\sqrt[3]{27}-14 \div 2)(6-8)^2$ is $-16$ .

In conclusion, simplifying mathematical expressions requires a systematic approach, and adhering to the order of operations (PEMDAS/BODMAS) is paramount. By breaking down the expression into smaller, manageable steps, we can navigate through the complexities and arrive at the correct answer. In this detailed guide, we meticulously simplified the expression $(\sqrt[3]{27}-14 \div 2)(6-8)^2$ , demonstrating each step with clarity and precision. From simplifying within parentheses to dealing with exponents and performing multiplication, we followed the PEMDAS/BODMAS guidelines to ensure accuracy. The final result, $-16$ , is a testament to the power of systematic problem-solving and the importance of understanding mathematical principles. Simplifying expressions is not just a mathematical exercise; it's a valuable skill that enhances our ability to approach complex problems in various aspects of life. By mastering the art of simplification, we equip ourselves with a powerful tool for critical thinking and decision-making. So, let's embrace the challenges that mathematical expressions present and continue our journey towards mathematical proficiency. The ability to simplify expressions is not just a mathematical skill; it's a gateway to a deeper understanding of mathematical concepts and their applications in the real world. It requires attention to detail, a firm grasp of mathematical principles, and a systematic approach. By following the steps outlined in this guide, you can confidently tackle similar simplification problems and enhance your mathematical abilities. Remember, mathematics is not just about numbers and equations; it's about logical reasoning and problem-solving. The process of simplification is a journey of discovery, where each step brings us closer to the final solution. So, let's continue to explore the world of mathematics with curiosity and enthusiasm.

Simplify the expression $(\sqrt[3]{27}-14 \div 2)(6-8)^2$ , showing each step.

Simplifying Expressions Step-by-Step Guide for $(\sqrt[3]{27}-14 \div 2)(6-8)^2$