Solving Math Expressions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a fun little problem that involves substituting values into an expression and then crunching the numbers. We'll be working with the expression: $\frac{1}{4}\left(c^3+d^2\right)$. The key here is to understand the order of operations and how to correctly substitute the given values for c and d. So, buckle up, and let's get started! This is a great opportunity to flex our algebra muscles and make sure we're comfortable with exponents and basic arithmetic. The best part? It's really not as scary as it might look at first glance. We'll break it down into easy-to-follow steps, so you'll be a pro in no time. We will begin with the substitution of values, then simplification by power operations, and lastly the final calculations. Let's make this fun! Mathematics is something we can learn by doing.

Understanding the Problem: The Core of the Math

The first step in tackling any math problem is to truly understand what's being asked. In this case, we have an algebraic expression: $\frac{1}{4}\left(c^3+d^2\right)$. This expression contains variables, c and d, and we're given specific values for them: c = -4 and d = 10. Our goal is to substitute these values into the expression and then calculate the final result. Think of it like a recipe. The expression is the recipe, and the values of c and d are the ingredients. We're going to put the ingredients (the numbers) into the recipe (the expression) and see what we get!

Before we begin, remember the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division - from left to right, Addition and Subtraction - from left to right). This order is crucial for solving mathematical expressions because it ensures that everyone arrives at the same answer. It tells us the sequence in which we must solve the expression. Without it, we would get totally different answers. In this problem, we must deal with parentheses, exponents, and then a division, so we will keep this in mind. It is also good to remember that, when dealing with negative numbers, we must take special care. It's really easy to make small mistakes that can completely change the answer, so keeping our focus is essential to be successful. That's why practicing with many examples is a great way to improve our skills and increase our ability to do it correctly. This ensures we don't overlook any crucial steps. This practice helps us build confidence and become more comfortable with these types of problems.

To summarize, we are going to start with an expression, we will then use a substitution with the given values, next we will simplify by doing power operations and finally, we will calculate the final value. It is also important to remember that, in this problem, we have a fraction. So, always remember that, when you have a fraction like this, the first thing to calculate is the numerator (everything inside the parentheses) and, finally, divide by the denominator (4 in this case). Alright, let's get down to business!

Step-by-Step Solution: Crunching the Numbers

Alright, let's roll up our sleeves and solve this math problem step-by-step. First, we need to substitute the given values of c and d into our expression: $\frac{1}{4}\left(c^3+d^2\right)$. We know that c = -4 and d = 10. So, we'll replace c with -4 and d with 10. Our expression now becomes: $\frac{1}{4}\left((-4)^3+(10)^2\right)$. See? It's that easy. Substitution is the key to unlock the problem.

Next, we're going to tackle the exponents. Remember, exponents mean we multiply a number by itself a certain number of times. Let's start with $(-4)^3$. This means -4 multiplied by itself three times: $(-4) * (-4) * (-4)$. A negative number multiplied by a negative number gives a positive number, and a positive number multiplied by a negative number gives a negative number. This means $(-4) * (-4) = 16$, and then $16 * (-4) = -64$. So, $(-4)^3 = -64$. Now, let's look at $(10)^2$. This means 10 multiplied by itself twice: $10 * 10 = 100$. So, $(10)^2 = 100$. Our expression now looks like this: $\frac{1}{4}\left(-64+100\right)$. We're doing great, guys! Keep going! This part of the process is very important, because if we make a mistake here, the final result will be wrong. That's why being very careful is essential, and always re-check our results, at least once. If possible, we can use a calculator to make sure we did the operations correctly.

Now, let's focus on what's inside the parentheses: $-64 + 100$. When we add these two numbers, we get 36. So our expression simplifies to: $\frac{1}{4}*(36)$. That is so awesome!

Finally, the last step is to divide 36 by 4: $\frac{36}{4} = 9$. And there you have it! The value of the expression when c = -4 and d = 10 is 9. Awesome! You've done it!

Breaking Down the Process: Key Takeaways

Let's recap what we've learned and highlight the key steps in solving this kind of problem. First, we have to start by understanding the problem and identifying the values. Then, we substitute the variables by the given values. After that, we must take care of the exponents. Then, simplify inside the parentheses and finally, divide or multiply as required. Pretty simple, right? The most important thing to remember is the order of operations (PEMDAS). This tells you the correct sequence to perform the calculations. Incorrect order will get you a wrong answer, so always be mindful of it. Practicing is key! The more you work with these types of problems, the more comfortable and confident you'll become. Each problem you solve is a step forward in strengthening your math skills. Don't be afraid to make mistakes; they are a part of the learning process. The great thing is that every time we make mistakes, we will learn more, which will help us avoid the same mistakes in the future. Embrace the challenge, enjoy the process, and celebrate your successes! Math can be a lot of fun if you approach it the right way. Keep practicing and keep learning, and you'll be amazed at what you can achieve. Also, don't forget to ask for help if you need it. There are lots of resources available to help you understand these math concepts. You can seek help from a teacher, classmates, or online resources. Learning is always easier when we do it together.

Conclusion: You've Got This!

Congratulations, guys! You've successfully solved the expression! You took the given expression, substituted the values, handled the exponents, simplified inside the parentheses, and then performed the final calculation. You have shown that you know the order of operations, a fundamental concept in mathematics. Remember, practice is key, and every problem you solve makes you better. So keep up the amazing work! Keep exploring, keep questioning, and keep learning. The world of mathematics is vast and fascinating, and you are well on your way to mastering it. Keep in mind that math is not just about getting the right answer; it's about the process of problem-solving, the critical thinking, and the ability to break down complex problems into smaller, manageable steps. These skills are invaluable, not just in math, but in all aspects of life. So, go out there and keep challenging yourself. You've got this!