Solve For P: A Math Problem Explained

Hey everyone, let's dive into a cool math problem! We're given a specific expression involving 'p' and our mission is to figure out the value of another expression that also involves 'p'. Don't worry, it's not as scary as it sounds. We'll break it down step by step, making sure everyone understands the process. This is a common type of algebra problem, and mastering it will definitely boost your math skills. So, grab your pencils, and let's get started. We're going to use the information given, solve for p first, and then plug that value into a new expression. Sound good?

This kind of problem is all about substitution and understanding how variables work. You'll often see these types of questions in algebra, calculus, and even in more advanced math topics. The core concept is about using given information to find the value of an unknown variable, and then using that value to calculate another expression. Being able to solve for a variable and substitute it into an expression is a fundamental skill in math. It’s like having a secret code that unlocks the solution to many problems. We are going to find a value of p, and then we are going to use that value to find the values of 3p³ + 9p. So, let's get down to the nitty-gritty and see how it works. First, we need to know what we have, and then we need to know what we are looking for, and finally, we need to connect the two.

We start with the information we are given, that is, p x1 3 x 1 3. It can be rewritten as p * x * 13 * x * 13*. However, there is no value of x given. Let us assume that this is not an expression, but an equation. If this were an equation, we might have something like p = x * 13 * x * 13*. And then we could solve for p. But since this is a problem and we need to find the value of something else, let us assume that x = 1. Therefore, our base value becomes: p * 1 * 13 * 1 * 13*. And we can rewrite that as p = 169. Now that we have the value for p we can move forward.

Finding the Value of 3p³ + 9p

Alright, now that we've found the value of p (which is 169 in our assumed case), we can finally get to the heart of the problem. Our next task is to calculate the value of the expression 3p³ + 9p. It's all about substituting the value we found for p into this new expression and then simplifying it. This part might look a bit intimidating, but trust me, it’s just a matter of careful calculation. Keep in mind the rules of exponents and order of operations (PEMDAS/BODMAS), and you’ll be golden. We'll carefully substitute the value of p into the expression, which is basically replacing every 'p' with '169'. Then, we'll follow the order of operations, first dealing with the exponent (the cube) and then doing the multiplication and addition. It's a simple process, but accuracy is key. Remember, in algebra, every step matters, and the more careful you are, the less likely you are to make mistakes. So, let’s do it!

To find the value of 3p³ + 9p, let’s substitute p with 169. This gives us 3 * (169)³ + 9 * (169). Now, let’s calculate (169)³. It is 169 * 169 * 169 which equals 4,862,739. Let's substitute that back in the equation. Now we have 3 * (4,862,739) + 9 * (169). Then multiply 3 by 4,862,739 which equals 14,588,217. Then, multiply 9 * 169, which equals 1,521. Now add these values 14,588,217 + 1,521. And there you have it, the answer is 14,589,738. Pretty cool, right? We took a somewhat complicated-looking expression and, with a bit of algebra and substitution, found its numerical value. This is a very powerful technique, and you can apply this to many problems. Remember that the more you practice these steps, the easier they become. Keep going, and you'll find that you can solve these types of problems with ease.

Now, let's take a look at the calculations:

1. Substitute: We know that p = 169, so our expression becomes: 3 * (169)³ + 9 * 169.
2. Calculate the Cube: (169)³ = 4,862,739.
3. Multiply: 3 * 4,862,739 = 14,588,217 and 9 * 169 = 1,521.
4. Add: 14,588,217 + 1,521 = 14,589,738.

So, the value of 3p³ + 9p is 14,589,738!

Why This Matters

Understanding how to solve these kinds of problems isn’t just about getting the right answer on a test. It’s about building a strong foundation in mathematics. Algebra, in particular, is a fundamental building block for many other areas of math, like calculus, which is used in so many different fields, from engineering to computer science. The ability to manipulate variables, substitute values, and simplify expressions is crucial for tackling more complex problems. Plus, this method of breaking down a complex problem into smaller, manageable steps is a useful skill that you can apply in other areas of life too! Think of it as a logical approach to problem-solving. This problem teaches you how to systematically approach a problem, break it down, and find a solution.

This same approach can be used in many real-world scenarios. Imagine you're trying to figure out the best way to manage your budget. You might use algebra to understand how changes in your income affect your expenses and savings. Or, if you're working on a project that involves different variables, like the speed of a car or the amount of time it takes to complete a task, you'll need to use these principles. So, by practicing these basic algebraic concepts, you're building a versatile skill set that will benefit you in all sorts of different ways, even when you least expect it!

Remember, practice makes perfect. The more you work through these types of problems, the more comfortable and confident you’ll become. And if you ever get stuck, don't hesitate to go back and review the steps, or look for extra examples. Math can be fun and rewarding, and it opens up a whole new world of possibilities. Keep up the great work!