Simplifying Math: A Student's Journey Through The Equation

Hey everyone! Today, we're diving into a math problem that a student tackled step-by-step. We'll break down the process of how to simplify $(\sqrt[3]{64}-16+2)(2-4)^2$, following the student's work. Get ready to flex those math muscles, because we're about to make this equation our friend!

Step-by-Step Breakdown of the Simplification Process

Let's start by listing the steps provided by the student, making sure we understand each move. It's like having a cheat sheet to guide us through the problem. We'll begin with the original equation and carefully go through each step to simplify it: $(\sqrt[3]{64}-16+2)(2-4)^2$. The goal here is to arrive at the correct solution while understanding every operation. Let’s not get lost; let's take this one step at a time! This method is awesome to clarify how to solve even the most challenging math problems. So, buckle up! You're about to learn how to master this type of problem. We will also focus on the core concepts here, such as the order of operations, the different mathematical operations, and how to apply them to solve any given problem. The more you practice, the easier it gets. It's like riding a bike: a little wobbly at first, but soon you’ll be cruising along with no problem.

Step 1: Simplifying the Cube Root

The first step involves simplifying the cube root. The student correctly identified that the cube root of 64 is 4. In mathematical terms, this means we're looking for a number that, when multiplied by itself three times, equals 64. That number is 4 (because 4 * 4 * 4 = 64). So, the student replaced $\sqrt[3]{64}$ with 4, transforming the equation to $(4-16+2)(2-4)^2$. This initial step is critical because it reduces the complexity of the equation, making the subsequent steps much easier to manage. Remember, understanding the basic operations is important to master these types of questions. Take your time, and don't rush through the initial steps.

This simple substitution is not just about getting the right answer; it's about setting up the problem to be solved easily. By reducing the complexity of the equation, you are essentially creating a smoother path to the solution. The core idea is to break the big problem into smaller, more manageable parts. This strategy is not only helpful in math but also in various other fields. So, embrace these initial steps, as they are the cornerstones of your problem-solving journey. It’s like setting the foundation of a building; if the foundation is not right, the whole structure will be unstable. That's why each step matters. Ensure you understand what is happening at each stage to avoid confusion later. The cube root step might seem basic, but it showcases the start of simplifying the expression. Make sure you're comfortable with these foundational concepts. Without them, it will be hard to progress. It's really that simple.

Step 2: Simplifying the Expression Inside the First Parentheses

Next, the student started to simplify the first set of parentheses. The equation $(4-16+2)(2-4)^2$ is now simplified within the parentheses. The student combined 4 and -16, then added 2. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates that we handle operations inside parentheses first. Thus, simplifying within the first set of parentheses leads to $(-12+2)(2-4)^2$.

This is a critical step because it reduces the number of terms and makes the equation clearer. By performing these operations, you're not just moving towards the answer but also enhancing your ability to deal with complex expressions. Many people find that math can be tricky, but understanding the rules will make everything easier. The student has carefully applied the rules of addition and subtraction to make sure each step is right. Keep in mind that accuracy is the name of the game. Math is all about precision. Every tiny detail matters. In this step, you are reducing the equation to its simplest form. That's why it is so important. Make sure you feel confident about each operation.

Step 3: Continuing the Simplification of the First Parentheses

Following step 2, the student continued to simplify the equation, resulting in the simplification of the expression inside the first set of parentheses. From $(-12+2)(2-4)^2$, the equation is further simplified. The student now combined -12 and 2, which equals -10. This results in the equation $(-10)(2-4)^2$. At this stage, the equation is becoming more manageable. Each step is designed to bring you closer to the final answer. Make sure you don't skip steps, as each one plays a part. You'll soon see how all the calculations come together. By taking the time to work through the details, you are not only solving the problem, but also building a solid foundation in mathematics. It's really that simple, just one step at a time, each calculation is bringing you closer to the end goal. Never underestimate the importance of taking your time, especially in math.

Step 4: Simplifying the Second Parentheses and Preparing for Exponentiation

In this step, the focus shifts to the second set of parentheses. The student dealt with $(2-4)^2$. The operations within the parentheses must be simplified. Therefore, the student subtracted 4 from 2 to obtain -2. The equation is now $(-10)(-2)^2$. It's a key step as you prepare for exponentiation, which follows the order of operations (PEMDAS). This step is necessary to get to the answer, and now the equation is much simpler. Keep track of the signs. It's easy to overlook them. This small step sets up the next operation, which will greatly affect the final answer. By correctly handling the parentheses, you're setting yourself up for success. So, take the time to simplify everything properly. Keep in mind that understanding and mastering the rules of math is crucial for succeeding. It’s like learning the rules of a game; if you know them well, you are more likely to win.

Step 5: Applying the Exponent

Now, the student addressed the exponent. The equation is $(-10)(-2)^2$. The exponent, which is 2, applies to -2. Squaring -2 means multiplying -2 by itself: $(-2) * (-2) = 4$. So, the equation is now $(-10)(4)$. This is the stage where the power of exponents is fully realized. Remember, a negative number squared becomes positive. If you remember that, you are golden. Exponents are not just about finding the answer; they also enhance the skills to deal with more advanced math. Understanding how exponents work is fundamental. This is like understanding how a car engine works; it will help you understand more complex mechanical systems.

It is important to understand what is happening in each step, and don’t be afraid to take a few extra steps if necessary. This process is about building a strong foundation in math, one step at a time. The exponent operation is often overlooked. But not anymore! Now you understand how it works. By understanding the exponent, you not only solve this problem, but you also build your math skills for future problems. With a solid grasp of exponents, you are one step closer to mastering more complex equations. The more you learn, the more confident you become. So, keep practicing!

Step 6: Completing the Calculation

In the final step, the student finishes the calculation. The equation is $(-10)(4)$. The student multiplies -10 by 4, which equals -40. Therefore, the solution to the original equation $(\sqrt[3]{64}-16+2)(2-4)^2$ is -40. This last step brings everything together. It's the culmination of all the previous steps. It's like putting the last piece of a puzzle together. Every calculation leads to this moment. The solution is not just an answer; it's the result of carefully following the order of operations and understanding the math rules. You’ve reached the final step, and you’ve got the correct answer. Give yourself a pat on the back!

This final step is your reward for all your effort. It proves that you've successfully navigated the equation. Each step you've learned has added to your math knowledge. Now you can solve equations of this type. You’ve demonstrated that you understand the different operations and the order in which they should be applied. Make sure you revisit this equation if you get stuck. Your success is proof that you have mastered the problem. You did it! Congratulations.

Conclusion: Mastering the Math Equation

So, guys, there you have it! We've meticulously followed the student's steps to simplify $(\sqrt[3]{64}-16+2)(2-4)^2$. By breaking down each step and explaining the reasoning behind it, we've transformed what might seem like a complicated equation into something much more approachable. Remember, math is like any other skill. The more you practice, the better you become. Every problem you solve adds to your knowledge and confidence. Embrace the process, don't be afraid to make mistakes (they're part of learning!), and always remember the order of operations: PEMDAS. Keep going, and you'll be amazed at what you can achieve. Math is a journey, not a destination. Enjoy the ride! And until next time, keep crunching those numbers!