Cube Volume: Step-by-Step Calculation

by Pedro Alvarez 38 views

Hey guys! Ever wondered how to calculate the volume of a cube, especially when the side length looks a bit complicated? Don't worry, it's not as daunting as it seems! In this guide, we'll break down the process step-by-step, using a specific example where the side length is expressed as (3/2a - 2/5n)³. We'll make sure it's super clear and easy to follow, so you can confidently tackle similar problems in the future. Let's dive in!

Understanding the Basics: Volume of a Cube

Before we get into the nitty-gritty, let's quickly recap the fundamentals of cube volume calculation. A cube, as you know, is a three-dimensional shape with six identical square faces. Think of a dice or a Rubik's Cube – those are perfect examples! The volume of any three-dimensional object tells us how much space it occupies. For a cube, this is super straightforward to calculate.

The formula for the volume (V) of a cube is simply the side length (s) cubed, which means raised to the power of 3. Mathematically, we write it as: V = s³. This means if you know the length of one side of the cube, you can easily find its volume by multiplying that length by itself three times (s * s * s). It's that simple! But what happens when the side length isn't just a simple number, but an expression like (3/2a - 2/5n)³? That's where things get a bit more interesting, and we need to use some algebraic skills to expand and simplify. In the next sections, we'll break down how to handle exactly that, so you'll be a pro at calculating cube volumes no matter how complex the side length looks. We'll go through each step meticulously, ensuring you understand the logic behind each operation. Remember, math isn't about memorizing formulas; it's about understanding the concepts and applying them confidently. So, let's get started and unravel the mystery of calculating the volume of a cube with a complex side!

The Challenge: Side Length of (3/2a - 2/5n)³

Okay, here’s where things get a little more interesting. We're not just dealing with a simple number for the side length; instead, we've got an expression: (3/2a - 2/5n)³. This means the side of our cube is represented by this algebraic expression, and we need to cube it to find the volume. Sounds intimidating? Don't worry, we'll break it down into manageable steps. The key here is to remember our algebraic rules, especially how to expand expressions raised to a power.

When you see something like (3/2a - 2/5n)³, it means we're multiplying the expression (3/2a - 2/5n) by itself three times: (3/2a - 2/5n) * (3/2a - 2/5n) * (3/2a - 2/5n). Now, you might be tempted to distribute the cube directly to each term inside the parentheses, but that's a common mistake! We can't simply say (3/2a)³ - (2/5n)³. Instead, we need to expand the expression step-by-step, usually by first multiplying two of the expressions together and then multiplying the result by the remaining expression. This might sound like a lot of work, but with a systematic approach, it becomes much clearer. We'll use techniques like the distributive property (or the FOIL method) to carefully multiply the terms. Think of it like building a house – you lay the foundation first, then the walls, and finally the roof. Similarly, we'll multiply the expressions in a structured way to avoid errors and ensure we get the correct final answer. So, buckle up, and let's get ready to expand this expression and find the volume of our cube! We'll start by tackling the first multiplication, making sure we're solid on each step before moving on. Remember, patience and accuracy are your best friends in algebra!

Step-by-Step Expansion: (3/2a - 2/5n)³

Alright, let's get our hands dirty and start the expansion. Remember, we're tackling (3/2a - 2/5n)³. As we discussed, the first step is to multiply (3/2a - 2/5n) by itself. This is where the distributive property (or the FOIL method, which stands for First, Outer, Inner, Last) comes into play. We'll multiply each term in the first expression by each term in the second expression.

So, let's break it down:

  1. (First): Multiply the first terms: (3/2a) * (3/2a) = 9/4 a²
  2. (Outer): Multiply the outer terms: (3/2a) * (-2/5n) = -6/10 an (which simplifies to -3/5 an)
  3. (Inner): Multiply the inner terms: (-2/5n) * (3/2a) = -6/10 an (which also simplifies to -3/5 an)
  4. (Last): Multiply the last terms: (-2/5n) * (-2/5n) = 4/25 n²

Now, let's combine these results: 9/4 a² - 3/5 an - 3/5 an + 4/25 n². We can simplify this further by combining the like terms (-3/5 an and -3/5 an). This gives us: 9/4 a² - 6/5 an + 4/25 n². Great! We've successfully multiplied (3/2a - 2/5n) by itself. But we're not done yet. Remember, we need to cube the expression, so we still need to multiply this result by (3/2a - 2/5n) one more time. This is where things can get a little more complex, but don't worry, we'll approach it systematically, just like we did in this step. We'll distribute each term in our new expression (9/4 a² - 6/5 an + 4/25 n²) by each term in (3/2a - 2/5n). It might seem lengthy, but breaking it down term by term will help us avoid mistakes. So, let's take a deep breath and get ready for the final multiplication! We're almost there, and the feeling of accomplishment when we get to the final answer will be totally worth it.

Final Multiplication and Simplification

Okay, guys, we're on the home stretch! We've already tackled the first part of the expansion, and now we need to multiply our result, 9/4 a² - 6/5 an + 4/25 n², by (3/2a - 2/5n). This is the final step in expanding (3/2a - 2/5n)³, and it involves carefully distributing each term.

Let's break it down systematically. We'll multiply each term in the first expression (9/4 a² - 6/5 an + 4/25 n²) by each term in the second expression (3/2a - 2/5n). This will give us a series of terms that we'll then need to simplify by combining like terms.

Here we go:

  1. (9/4 a²) * (3/2a) = 27/8 a³
  2. (9/4 a²) * (-2/5n) = -18/20 a²n (which simplifies to -9/10 a²n)
  3. (-6/5 an) * (3/2a) = -18/10 a²n (which simplifies to -9/5 a²n)
  4. (-6/5 an) * (-2/5n) = 12/25 an²
  5. (4/25 n²) * (3/2a) = 12/50 an² (which simplifies to 6/25 an²)
  6. (4/25 n²) * (-2/5n) = -8/125 n³

Now, let's write out all the terms we've got: 27/8 a³ - 9/10 a²n - 9/5 a²n + 12/25 an² + 6/25 an² - 8/125 n³. It looks a bit messy, but the next step is to combine the like terms. We have a couple of terms with a²n and a couple with an². Let's combine them:

  • a²n terms: -9/10 a²n - 9/5 a²n. To combine these, we need a common denominator, which is 10. So, -9/10 a²n - 18/10 a²n = -27/10 a²n
  • an² terms: 12/25 an² + 6/25 an² = 18/25 an²

Now, let's rewrite the entire expression with the simplified terms: 27/8 a³ - 27/10 a²n + 18/25 an² - 8/125 n³. And there you have it! This is the expanded and simplified form of (3/2a - 2/5n)³, which represents the volume of our cube.

The Final Answer: Volume of the Cube

Alright, after all that careful expansion and simplification, we've arrived at our final answer! The volume (V) of the cube, where the side length is (3/2a - 2/5n)³, is:

V = 27/8 a³ - 27/10 a²n + 18/25 an² - 8/125 n³

Woohoo! Give yourselves a pat on the back, guys. That was a challenging problem, but we tackled it step-by-step and emerged victorious. You now know how to calculate the volume of a cube even when the side length is a complex expression. This is a valuable skill in algebra and geometry, and you can apply this knowledge to solve a wide range of problems.

Let's just take a moment to appreciate what we've accomplished. We started with a seemingly complicated expression, (3/2a - 2/5n)³, and we systematically broke it down using algebraic principles. We remembered the formula for the volume of a cube (V = s³), and we carefully expanded the expression, ensuring we didn't fall into the trap of distributing the cube directly. We used the distributive property (or FOIL method) to multiply the terms, and we meticulously combined like terms to simplify our answer. The result is a polynomial expression that represents the volume of the cube in terms of 'a' and 'n'.

This exercise wasn't just about getting the right answer; it was about the process. It's about learning how to approach complex problems, breaking them into smaller, manageable steps, and applying the tools and techniques you've learned. So, next time you encounter a similar challenge, remember this journey. Remember the systematic approach, the careful calculations, and the triumphant feeling of arriving at the final answer. You've got this!

Key Takeaways and Practice

So, what are the key things we've learned in this guide? Let's recap the essential takeaways to solidify your understanding:

  • Volume of a Cube: The volume of a cube is calculated by cubing the side length (V = s³).
  • Expanding Complex Expressions: When dealing with expressions like (3/2a - 2/5n)³, you can't simply distribute the exponent. Instead, you need to expand the expression step-by-step.
  • Distributive Property (FOIL): The distributive property (or FOIL method) is crucial for multiplying expressions with multiple terms.
  • Combining Like Terms: After expanding, always simplify your expression by combining like terms.
  • Systematic Approach: Break down complex problems into smaller, manageable steps to avoid errors and ensure accuracy.

Now that we've covered the theory and worked through an example, the best way to truly master this skill is through practice. Try tackling similar problems with different expressions for the side length. You could change the fractions, add more terms, or even introduce different variables. The more you practice, the more confident you'll become in your ability to expand and simplify algebraic expressions.

For example, you could try calculating the volume of a cube with side lengths like (a + b)³, (2x - y)³, or even more complex expressions like (1/2p + 3/4q)³. The process will be the same – expand the expression step-by-step, use the distributive property, and combine like terms. Don't be afraid to make mistakes; they're a natural part of the learning process. The key is to learn from your mistakes and keep practicing. You can also check your answers using online calculators or ask a teacher or tutor for help if you get stuck.

Remember, math is like a muscle – the more you exercise it, the stronger it becomes. So, keep practicing, keep exploring, and keep challenging yourself. You've got the tools and the knowledge to succeed!