Solving $(5x^3)(4x)^3$ A Step-by-Step Guide

Hey guys, ever stumbled upon a math problem that looks like it's speaking a different language? Well, today, we're going to crack the code of one such expression. We're diving into the world of exponents and algebraic manipulation to figure out which expression is equivalent to $(5x^3)(4x)^3$. Trust me, it's not as daunting as it looks! We'll break it down step by step, making sure everyone, from math newbies to seasoned pros, can follow along. So, buckle up and let's get started on this mathematical adventure!

Deciphering the Expression: A Step-by-Step Guide

When we're faced with an algebraic expression like $(5x^3)(4x)^3$, the key is to remember our fundamental rules of exponents and distribution. It's like having a secret decoder ring for math! Let's start by focusing on the second part of the expression, $(4x)^3$. This is where the power of a product rule comes into play. This rule states that $(ab)^n = a^n b^n$. In simpler terms, when you have a product raised to a power, you can distribute that power to each factor inside the parentheses. So, $(4x)^3$ becomes $4^3 * x^3$, which simplifies to $64x^3$. See? We're already making progress!

Now, let's bring back the first part of our expression, $(5x^3)$. Our expression now looks like this: $(5x^3)(64x^3)$. The next step involves the multiplication of like terms. We multiply the coefficients (the numbers in front of the variables) and add the exponents of the variables with the same base. So, we multiply 5 and 64, which gives us 320. Then, we multiply $x^3$ by $x^3$. Remember the rule of exponents that says $x^m * x^n = x^(m+n)$? Applying this rule, $x^3 * x^3$ becomes $x^(3+3)$, which simplifies to $x^6$. Putting it all together, we have $320x^6$. And that's it! We've successfully deciphered the expression. The equivalent expression to $(5x^3)(4x)^3$ is $320x^6$. So, the correct answer is B. $320x^6$.

Why This Matters: Real-World Applications of Exponents

Okay, so we've solved the problem, but you might be thinking, “When am I ever going to use this in real life?” That's a fair question! While you might not be simplifying algebraic expressions every day, the concepts behind them are incredibly important in various fields. Exponents, in particular, are used extensively in science, engineering, and computer science. For instance, in physics, exponents are used to describe quantities that vary greatly in magnitude, such as the intensity of light or the strength of an earthquake. The Richter scale, which measures the magnitude of earthquakes, is a logarithmic scale, meaning it uses exponents to represent the size of the earthquake. An earthquake of magnitude 6 is ten times stronger than an earthquake of magnitude 5.

In computer science, exponents are fundamental to understanding data storage and processing speeds. Computers use binary code, which is based on powers of 2. The amount of memory in your computer or smartphone is often measured in gigabytes (GB), which are units of information based on powers of 2. Similarly, the speed of a computer processor is often measured in gigahertz (GHz), which represents billions of cycles per second. These units rely on exponential notation to express large numbers concisely. In finance, exponents are crucial for calculating compound interest. When you invest money, the interest you earn can also earn interest, leading to exponential growth of your investment. The formula for compound interest involves raising the interest rate to the power of the number of compounding periods. This concept is also important in understanding loans and mortgages, where interest is calculated on the outstanding balance. Understanding exponents can help you make informed decisions about your finances.

Common Pitfalls and How to Avoid Them

When working with exponents and algebraic expressions, it's easy to make mistakes if you're not careful. Let's talk about some common pitfalls and how to steer clear of them. One frequent error is forgetting to distribute the exponent to all factors within the parentheses. For instance, in our problem, some might incorrectly calculate $(4x)^3$ as $4x^3$ instead of $4^3x^3$ or $64x^3$. Always remember that the exponent applies to everything inside the parentheses. Another common mistake is with negative signs. Remember that a negative number raised to an even power becomes positive, while a negative number raised to an odd power remains negative. For example, $(-2)^2 = 4$, but $(-2)^3 = -8$. Pay close attention to the signs when performing calculations.

Another area where errors often occur is in applying the correct order of operations. Remember the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). Always perform operations in the correct order to avoid mistakes. For example, in the expression $2 + 3 * 2^2$, you should first calculate $2^2$ (which is 4), then multiply 3 by 4 (which is 12), and finally add 2, giving you a result of 14. If you perform the operations in the wrong order, you'll get a different answer. Lastly, avoid the temptation to add exponents when multiplying terms with the same base. Remember, when multiplying terms with the same base, you add the exponents. However, when raising a power to a power, you multiply the exponents. For instance, $x^2 * x^3 = x^(2+3) = x^5$, but $(x^2)^3 = x^(2*3) = x^6$. Keeping these rules straight will help you avoid many common errors. By being mindful of these common pitfalls and practicing regularly, you can improve your accuracy and confidence in working with exponents and algebraic expressions.

Practice Makes Perfect: Try These Problems

Alright, now that we've dissected the problem and discussed common pitfalls, it's time to put your knowledge to the test! The best way to master exponents and algebraic expressions is through practice. So, let's try a few more problems. Grab a pencil and paper, and let's get to work!

Problem 1: Simplify the expression $(3y^2)(2y)^4$.

Take a moment to work through this problem. Remember to apply the power of a product rule and the rule for multiplying terms with the same base. Distribute the exponent, multiply the coefficients, and add the exponents. Don't rush; take your time and double-check your work. The solution to this problem is $48y^6$.

Problem 2: Which expression is equivalent to $(a^2b^3)^2(ab^2)$?

This problem combines multiple exponent rules. First, you need to apply the power of a power rule, which states that $(x^m)^n = x^(m*n)$. Then, you need to multiply the resulting expressions, remembering to add the exponents of like variables. Break the problem down into smaller steps, and you'll find the solution. The correct answer is $a^5b^8$.

Problem 3: Simplify $\frac{12x^5}{4x^2}$.

This problem introduces division of terms with exponents. When dividing terms with the same base, you subtract the exponents. Divide the coefficients and then subtract the exponents of the variables. Remember the rule: $x^m / x^n = x^(m-n)$. The simplified expression is $3x^3$.

By working through these practice problems, you're reinforcing your understanding of exponent rules and building your problem-solving skills. The more you practice, the more comfortable and confident you'll become with these concepts. So, keep practicing, and you'll be an exponent expert in no time!

Wrapping Up: Key Takeaways and Final Thoughts

We've journeyed through the world of exponents and algebraic expressions, and what a trip it's been! We started with a seemingly complex problem, $(5x^3)(4x)^3$, and broke it down into manageable steps. We learned about the power of a product rule, the multiplication of like terms, and the importance of the order of operations. We also explored real-world applications of exponents and discussed common pitfalls to avoid. By practicing, we've sharpened our skills and built our confidence in tackling similar problems. The key takeaway here is that even the most challenging problems can be solved by breaking them down into smaller parts and applying the fundamental rules. Remember the exponent rules, be mindful of the order of operations, and practice regularly. With these tools in your arsenal, you'll be well-equipped to handle any algebraic challenge that comes your way. So, keep exploring, keep practicing, and never stop learning! Math can be an exciting adventure, and every problem you solve is a step forward on your mathematical journey.