Expressions Equivalent To (4x^5)(5x^6) A Detailed Guide

Hey everyone! Today, we're diving into the fascinating world of algebraic expressions, and we're going to unravel the mystery behind finding expressions that are equivalent to a given one. Our main focus is on the expression (4x⁵)(5x⁶). We'll break down the process step by step, ensuring you grasp the core concepts and can confidently tackle similar problems. This isn't just about getting the right answer; it's about understanding why the answer is correct. Think of it as building a strong foundation in algebra, brick by brick. So, grab your metaphorical hard hats, and let's get to work!

Understanding the Basics: Multiplying Expressions

Before we jump into the specific problem, let's quickly review the fundamental principles of multiplying algebraic expressions, guys. Remember, when we multiply terms with the same base (in this case, 'x'), we add their exponents. This is a crucial rule, often called the product of powers rule, and it's the cornerstone of what we're doing today. So, if you have xᵃ * xᵇ, it simplifies to xᵃ⁺ᵇ. Don't forget this; it's your superpower for this challenge! Also, remember that coefficients (the numbers in front of the variables) are multiplied directly. So, 4x⁵ * 5x⁶ becomes (4 * 5) * (x⁵ * x⁶). See how we've separated the coefficients and the variables? This makes the process much clearer. Understanding this separation is key to avoiding common mistakes. Think of it like organizing your tools before starting a project – it makes everything run smoother. Now, with this foundation, we can confidently tackle our problem.

Let's apply this to our initial expression, (4x⁵)(5x⁶). We multiply the coefficients (4 and 5) to get 20. Then, we multiply the variables (x⁵ and x⁶). Using the product of powers rule, we add the exponents (5 and 6) to get x¹¹. So, (4x⁵)(5x⁶) simplifies to 20x¹¹. This is our target expression. Now, the real challenge begins: identifying which of the given options are equivalent to this. It's like being a detective and looking for clues that match our suspect – 20x¹¹. We need to carefully examine each option and see if it simplifies to the same expression. This involves applying the same rules of multiplication and exponent manipulation that we just discussed. So, let's put on our detective hats and start investigating!

Evaluating the Candidate Expressions

Now, let's carefully examine each of the candidate expressions to see if they match our target expression, 20x¹¹. This is where the rubber meets the road, guys. We'll take each option one by one, simplify it using the rules we discussed earlier, and then compare the result to 20x¹¹. It's like a scientific experiment – we have a hypothesis (the option is equivalent) and we need to test it rigorously. This step-by-step approach is crucial for accuracy and helps us avoid overlooking any subtle differences. Remember, attention to detail is key in algebra. A small mistake in arithmetic or exponent manipulation can lead to a completely wrong answer. So, let's proceed with precision and care.

Option 1: (2x⁵)(10x⁶)

Let's start with the first option: (2x⁵)(10x⁶). To determine if it's equivalent, we need to multiply the coefficients and the variables separately. The coefficients are 2 and 10, and their product is 2 * 10 = 20. That's a good start! Now let's look at the variables. We have x⁵ and x⁶. Multiplying these using the product of powers rule, we add the exponents: 5 + 6 = 11. So, we get x¹¹. Putting it all together, (2x⁵)(10x⁶) simplifies to 20x¹¹. Bingo! This expression matches our target expression, 20x¹¹. So, this one is definitely equivalent. But we can't stop here; we need to evaluate all the options to be sure we've identified all the correct ones. Think of it like completing a puzzle – you don't stop after finding one piece that fits; you keep going until the whole puzzle is solved.

Option 2: (4x⁵)(6x⁵)

Next up is (4x⁵)(6x⁵). Let's follow the same process. Multiplying the coefficients, we have 4 * 6 = 24. Uh oh, that's already different from our target expression, which has a coefficient of 20. But let's continue the process just to be thorough. Now, let's multiply the variables: x⁵ * x⁵. Adding the exponents, we get 5 + 5 = 10. So, we have x¹⁰. Putting it together, (4x⁵)(6x⁵) simplifies to 24x¹⁰. This is clearly not equivalent to 20x¹¹ because both the coefficient and the exponent are different. So, we can confidently eliminate this option. It's like finding a fingerprint at a crime scene that doesn't match our suspect – it helps us narrow down the possibilities. This option serves as a good reminder of the importance of checking both the coefficient and the exponent when determining equivalence.

Option 3: (4x⁶)(5x⁵)

Finally, let's examine the third option: (4x⁶)(5x⁵). This one might look tricky at first glance, but let's break it down. Multiplying the coefficients, we have 4 * 5 = 20. Great! The coefficient matches our target expression. Now, let's multiply the variables: x⁶ * x⁵. Adding the exponents, we get 6 + 5 = 11. So, we have x¹¹. Putting it all together, (4x⁶)(5x⁵) simplifies to 20x¹¹. This expression is also equivalent to our target expression, 20x¹¹. Notice that the order of the terms with exponents doesn't matter because multiplication is commutative (a * b = b * a). This is a crucial concept to remember and can often be a source of confusion if overlooked. So, we've identified another equivalent expression.

Conclusion: Identifying Equivalent Expressions

Alright guys, we've reached the finish line! After carefully evaluating each option, we've successfully identified the expressions equivalent to (4x⁵)(5x⁶). Remember, our target expression was 20x¹¹. We found that (2x⁵)(10x⁶) and (4x⁶)(5x⁵) both simplify to 20x¹¹, making them equivalent. The expression (4x⁵)(6x⁵), however, simplifies to 24x¹⁰, which is not equivalent. This exercise highlights the importance of carefully applying the rules of exponents and multiplication when simplifying algebraic expressions.

This wasn't just about finding the right answers; it was about understanding the process. We reviewed the product of powers rule, the importance of multiplying coefficients and variables separately, and the commutative property of multiplication. These are fundamental concepts that will serve you well as you continue your algebraic journey. Keep practicing, guys, and remember that every problem is an opportunity to strengthen your understanding. So, go forth and conquer those algebraic expressions with confidence!