Evaluate 5x^3 For X=2: A Step-by-Step Guide

Hey guys! Today, we're diving into a fun little math problem where we need to evaluate an algebraic expression. Don't worry, it's not as intimidating as it sounds! We're going to break it down step-by-step so everyone can follow along. Our mission, should we choose to accept it (and we do!), is to figure out the value of the expression 5x^3 when x is equal to 2. Sounds like a plan? Let's get started!

Understanding the Expression

Before we jump into plugging in numbers, let's take a moment to really understand what this expression, 5x^3, is telling us. In the world of algebra, expressions are like little puzzles made up of numbers, variables (like our 'x'), and operations (like the little '3' hanging up there, which is an exponent). So, what does each part mean in this particular expression?

First up, we have the number 5. This is what we call a coefficient. It's simply a number that's multiplying the rest of the expression. Think of it as five times whatever comes next.

Next, we have our variable, x. Variables are like placeholders. They can stand in for any number, and in this case, we're told that x is going to be 2. So, we'll be swapping out 'x' for '2' soon enough.

And finally, we have that little number sitting up high – the exponent, 3. This little guy tells us how many times we need to multiply the base (which is 'x' in this case) by itself. So, x^3 really means x * x * x. It's super important to remember the order of operations here. Exponents come before multiplication, so we need to deal with that exponent before we multiply by 5.

So, to recap, 5x^3 means: first, we take the value of x and multiply it by itself three times (because of the exponent). Then, we take that whole result and multiply it by 5. Understanding this is key to solving the problem correctly, so take a deep breath and make sure you're feeling good about it before we move on. Once you grasp this, the rest is a piece of cake!

Substituting the Value of x

Alright, now for the fun part! We know that x is equal to 2, and we need to find the value of 5x^3. So, what's the next logical step? That's right, we substitute! Substituting simply means replacing the variable 'x' in our expression with its given value, which is 2. It's like swapping out a piece in a puzzle to see if it fits.

So, let's do it. We take our expression, 5x^3, and wherever we see an 'x', we replace it with '2'. Now, it's super important to use parentheses when we substitute, especially when we're dealing with exponents. This helps us keep things organized and avoid any confusion. So, after substitution, our expression looks like this: 5(2)^3. See how we put the 2 inside parentheses? This tells us that the exponent 3 only applies to the 2, not to the 5. Without the parentheses, it could look like we're multiplying 5 by 2 first, which is definitely not what we want to do!

Now, let's think about why this substitution is so important. By replacing the variable with its actual value, we've transformed our algebraic expression into a simple arithmetic problem that we can easily solve. We've taken something that looked a bit abstract and turned it into something concrete. This is a fundamental skill in algebra, and you'll be using it all the time as you tackle more complex problems. So, make sure you're comfortable with the idea of substitution – it's your secret weapon for simplifying expressions!

Evaluating the Exponent

Okay, we've substituted x with 2, and now we have 5(2)^3. What's the next step? Remember our order of operations – PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). We've already dealt with the substitution, which you could think of as handling the parentheses. So, next up is the Exponent!

The exponent in our expression is 3, and it's sitting on top of the 2. This means we need to calculate 2 raised to the power of 3, or 2 cubed. As we discussed earlier, 2^3 means 2 * 2 * 2. Let's break that down:

• 2 * 2 = 4
• 4 * 2 = 8

So, 2^3 is equal to 8. Now we can replace (2)^3 in our expression with 8. This gives us 5 * 8. See how we're making progress? We're slowly but surely simplifying the expression, one step at a time. Exponents might seem a little scary at first, but once you understand what they mean, they're really not that bad. They're just a shorthand way of writing repeated multiplication. And mastering exponents is crucial for all sorts of math, from algebra to calculus, so you're building a solid foundation here.

Remember, the key to evaluating exponents correctly is to take your time and be careful with your multiplication. It's easy to make a small mistake, especially when you're dealing with larger exponents, so double-check your work. And if you're ever unsure, don't hesitate to write out the multiplication explicitly, like we did here (2 * 2 * 2). It can really help to visualize what's going on and prevent errors.

Performing the Multiplication

We're almost there, guys! We've simplified our expression to 5 * 8. Now, we just have one simple operation left: multiplication. This is the home stretch, so let's bring it home strong!

What is 5 multiplied by 8? If you know your times tables, you'll know that 5 * 8 = 40. If you're not quite as confident with your multiplication facts, you can think of it as adding 5 eight times (5 + 5 + 5 + 5 + 5 + 5 + 5 + 5) or adding 8 five times (8 + 8 + 8 + 8 + 8). Either way, you'll arrive at the same answer: 40.

So, we've done it! We've successfully evaluated the expression 5x^3 when x is equal to 2. The final answer is 40. Pat yourselves on the back – you've tackled an algebraic problem and come out on top! This final step of multiplication is a great reminder of how important those basic arithmetic skills are. Even when we're dealing with more advanced concepts like exponents and variables, we still need to be solid on our fundamental operations. A strong foundation in arithmetic will make algebra (and all higher math) much, much easier.

Final Answer

So, after carefully substituting, evaluating the exponent, and performing the final multiplication, we've reached our destination! The value of the expression 5x^3 when x = 2 is 40. Yay! Give yourselves a round of applause, guys! You've successfully navigated this algebraic adventure. We took an expression with a variable and an exponent, and by following the order of operations and breaking things down step by step, we arrived at a clear, concrete answer. And that, my friends, is the power of mathematics!

But more than just getting the right answer, hopefully you've also gained a deeper understanding of the process involved. It's not just about plugging in numbers and churning out results; it's about understanding what each part of the expression means, why we perform the operations in a certain order, and how each step builds upon the last. This kind of understanding is what will truly make you a math whiz!

And remember, practice makes perfect. The more you work with expressions and equations, the more comfortable you'll become with them. So, don't be afraid to tackle new problems, make mistakes (we all do!), and learn from them. Keep exploring the fascinating world of math, and you'll be amazed at what you can achieve. Keep up the awesome work!