Solve The Math Expression: A Step-by-Step Guide
Hey guys! Let's dive into solving this math problem. We're given an expression and asked to find its value when we plug in specific values for the variables. It's like a fun little puzzle, and I'll walk you through it step by step. We'll start with the basics, break down the expression, substitute the values, and then carefully calculate the final answer. This is a common type of problem in algebra, and understanding it will help you with more complex equations later on. So, grab your calculators (or your brains!) and let's get started. The core concept here revolves around the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). We'll keep this in mind as we work through each part of the problem. This problem helps build a strong foundation in algebraic manipulation. Ready? Let's go!
Understanding the Problem: The Math Expression
Okay, so the main deal is this: we need to find the value of the expression $ rac{1}{4} imes extstyle(c3+d2)$, given that c = -4 and d = 10. The expression involves a few different mathematical operations: exponents, addition, and multiplication. Before we start plugging in numbers, let's break down the different parts. We've got c cubed (c^3), which means c multiplied by itself three times (c * c * c). Then we have d squared (d^2), which means d multiplied by itself twice (d * d). These values are then added together, and finally, the sum is multiplied by one-fourth (1/4). Always remember the significance of variables in mathematical expressions. They represent unknown values that can change, making the expression's outcome dependent on the specific values we assign to them. It's like a recipe where the final dish depends on the ingredients you choose and how you combine them. The key is to correctly substitute the values and follow the order of operations to arrive at the correct answer. The more we practice these types of problems, the easier it becomes.
Let’s make sure we're on the same page. Remember that the correct application of the order of operations is crucial. If we calculate the parts in the wrong order, we will end up with the wrong final answer! This might seem intimidating, but believe me, it's not. By breaking down the expression into smaller parts, we can tackle it step by step. This approach is not only useful for this problem, but is a fundamental problem-solving skill, applicable across various areas of mathematics and life. So, pay close attention to each step, and you'll be able to solve similar problems with confidence in the future. Now, let’s get into the nitty-gritty of it.
Step-by-Step Solution: Plugging in the Numbers
Now, let's put our variables to the test, and start replacing c and d with their given values of -4 and 10, respectively. The expression becomes $ rac1}{4} imes extstyle((-4)3+(10)2)$. Notice how I’ve used parentheses to keep things organized and to ensure that we correctly handle the negative sign when dealing with c. So, our first step is to calculate the exponent values. We start with the value of $c^3$, which is $(-4)^3$. This means we multiply -4 by itself three times{4} imes extstyle(-64 + 100)$. Adding these two terms gives us 36, so our expression becomes $ rac{1}{4} imes extstyle(36)$. This is where the multiplication by one-fourth comes in! It’s the same as dividing by four. Take your time, focus, and follow each step carefully. These problems seem tricky at first, but break them down step by step, and it becomes much easier. Ready to keep going? Let’s finish this thing!
Final Calculation: The Answer
So, we are now down to the final stretch! We've got $ rac{1}{4} imes extstyle(36)$. The last operation is a simple multiplication. Multiplying 36 by one-fourth (or dividing 36 by 4) gives us 9. Therefore, the value of the expression $ rac{1}{4} imes extstyle(c3+d2)$ when c = -4 and d = 10 is 9. This completes the problem, and we've successfully found the answer. We can confidently say that B. 9 is the correct answer! Give yourself a high-five! You’ve successfully worked through the problem, calculated all the values correctly, and arrived at the right solution. You have demonstrated a strong grasp of both the order of operations and the substitution of values into algebraic expressions. Remember, practice is key, and with each problem you solve, you'll feel more confident in your math abilities. Keep up the awesome work, and keep exploring the amazing world of mathematics! It might seem tough at times, but with persistence, you can conquer any challenge. Never hesitate to revisit any steps that may be unclear, and make sure you understand the reasoning behind each operation. Keep practicing, keep learning, and before you know it, you will be solving these types of problems with ease. Math is not just a set of rules, it's a powerful tool that helps us understand and interact with the world around us. Good job!