Equivalent Expression To (5x^3)(4x)^3: Math Guide

by Andrew McMorgan 50 views

Hey Plastik Magazine readers! Ever found yourself scratching your head over an algebra problem? Don't worry, we've all been there. Today, we're going to break down a common type of math question: simplifying expressions with exponents. Let's dive into a problem that looks a bit intimidating but is actually quite manageable once you understand the rules. We're tackling the expression (5x3)(4x)3, and our mission is to find out which of the given options (A. 20x^6, B. 320x^6, C. 500x^6, D. 8,000x^6) is the equivalent simplified form. So, grab your thinking caps, and let’s get started!

Understanding the Problem: (5x3)(4x)3

Okay, so we're faced with the expression (5x3)(4x)3. At first glance, it might seem like a jumble of numbers and letters, but don't fret! The key to cracking this problem is understanding the order of operations and the rules of exponents. Remember PEMDAS/BODMAS? Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This handy acronym tells us the sequence in which we should perform mathematical operations.

In this expression, we have parentheses and exponents, so we'll tackle those first. The term (4x)^3 means that everything inside the parentheses is being raised to the power of 3. This is where the power of a product rule comes into play. The power of a product rule states that (ab)^n = a^n * b^n. In simpler terms, if you have a product raised to a power, you can raise each factor in the product to that power. So, let's apply this rule to our expression. (4x)^3 becomes 4^3 * x^3. Now, we know that 4^3 is 4 * 4 * 4, which equals 64. So, (4x)^3 simplifies to 64x^3. See? We're already making progress!

Now, let's bring the rest of the expression back into the mix. We originally had (5x3)(4x)3, and we've just simplified (4x)^3 to 64x^3. So, our expression now looks like this: (5x3)(64x3). We're now dealing with multiplication, and the next rule we need is the product of powers rule. This rule states that when you multiply terms with the same base, you add their exponents. Mathematically, it looks like this: a^m * a^n = a^(m+n). In our case, we have x^3 multiplied by x^3. So, according to the rule, we add the exponents: 3 + 3 = 6. This means x^3 * x^3 = x^6.

But wait, we're not quite done yet! We also need to multiply the coefficients, which are the numbers in front of the variables. In our expression (5x3)(64x3), the coefficients are 5 and 64. So, we multiply 5 by 64. If you do the math, you'll find that 5 * 64 = 320. Now we have all the pieces of the puzzle. We've got our coefficient (320) and our variable with its exponent (x^6). Putting it all together, the simplified expression is 320x^6. And there you have it! We've successfully navigated the world of exponents and simplified our expression.

Step-by-Step Solution

Let's break down the solution step-by-step to make it super clear for everyone. Understanding each step is crucial, so you can apply these concepts to similar problems in the future. We'll go through each operation, explaining the 'why' behind the 'what'.

  1. Original Expression: We start with the expression (5x3)(4x)3. This is our starting point, and we need to simplify it to one of the given answer choices.
  2. Apply the Power of a Product Rule: The first thing we need to tackle is the term (4x)^3. Remember, the power of a product rule says that (ab)^n = a^n * b^n. So, we apply this rule to (4x)^3, which becomes 4^3 * x^3.
  3. Calculate 4^3: Now, let's simplify 4^3. This means 4 * 4 * 4, which equals 64. So, our term (4x)^3 is now simplified to 64x^3.
  4. Rewrite the Expression: We can now rewrite our original expression, replacing (4x)^3 with its simplified form: (5x3)(64x3).
  5. Apply the Product of Powers Rule: Next up is the product of powers rule, which states that a^m * a^n = a^(m+n). We have x^3 multiplied by x^3, so we add the exponents: 3 + 3 = 6. This gives us x^6.
  6. Multiply the Coefficients: Don't forget about the numbers in front of the variables! We have 5 and 64. Multiply them together: 5 * 64 = 320.
  7. Final Simplified Expression: Now, we combine the coefficient and the variable with its exponent: 320x^6.

So, after following these steps, we arrive at the simplified expression: 320x^6. This step-by-step breakdown should give you a clear understanding of how we got to the answer. Each rule we applied has a specific purpose, and understanding these rules is key to mastering algebra.

Common Mistakes to Avoid

When working with exponents and algebraic expressions, it's easy to stumble into common pitfalls. But don't worry, we're here to help you steer clear of these mistakes. Recognizing these errors is half the battle, and once you're aware of them, you'll be much more likely to avoid them. Let's take a look at some common mistakes and how to dodge them.

One frequent error is misapplying the power of a product rule. Remember, this rule states that (ab)^n = a^n * b^n. A common mistake is to only apply the exponent to one factor inside the parentheses, forgetting to apply it to all factors. For example, in our expression (4x)^3, some might incorrectly calculate this as 4 * x^3, forgetting to raise the 4 to the power of 3 as well. Always remember to distribute the exponent to every factor within the parentheses. This is super important for getting the correct answer. So, keep this in mind and double-check your work to ensure you're applying the rule correctly.

Another common mistake happens when using the product of powers rule. This rule, which states that a^m * a^n = a^(m+n), is crucial for simplifying expressions. The error often occurs when people multiply the exponents instead of adding them. For instance, when simplifying x^3 * x^3, someone might incorrectly calculate this as x^(3*3) = x^9 instead of the correct x^(3+3) = x^6. Remember, when multiplying terms with the same base, you add the exponents, not multiply them. This is a key distinction, so make sure you're adding those exponents!

Forgetting to multiply the coefficients is another slip-up that's easy to make. When you have an expression like (5x3)(64x3), it's essential to multiply the coefficients (the numbers in front of the variables) as well as dealing with the exponents. Some people might correctly simplify the variable part to x^6 but forget to multiply 5 by 64. Always remember to multiply the coefficients together. In this case, 5 * 64 gives us 320, which is a crucial part of the final answer.

Finally, a general mistake to watch out for is not following the order of operations (PEMDAS/BODMAS). This acronym reminds us to perform operations in the correct order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Skipping or changing the order can lead to incorrect simplifications. So, always double-check that you're following PEMDAS/BODMAS to ensure accurate calculations. By being mindful of these common mistakes, you'll be well-equipped to tackle similar algebra problems with confidence!

Why This Matters: Real-World Applications

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