Equivalent Expression For 2^(3x-4): A Step-by-Step Guide

by Kenji Nakamura 57 views

Hey there, math enthusiasts! Ever stumbled upon an exponential expression and felt a little lost in the woods? Don't worry, we've all been there. Today, we're going to break down a classic problem: finding an equivalent expression for 2^(3x-4). We'll explore the fundamental concepts, walk through the solution step-by-step, and even throw in some extra tips and tricks to help you master these types of problems. So, grab your thinking caps, and let's dive in!

Understanding Exponential Expressions

Before we tackle the main problem, let's make sure we're all on the same page with exponential expressions. In a nutshell, an exponential expression is a mathematical expression that involves a base raised to a power (also called an exponent). The general form looks like this: b^x, where 'b' is the base and 'x' is the exponent.

The exponent tells us how many times to multiply the base by itself. For example, 2^3 means 2 * 2 * 2, which equals 8. Easy peasy, right? But things can get a bit trickier when we start throwing in variables and more complex exponents, like in our expression 2^(3x-4). That's where the rules of exponents come in handy.

The Power of Exponent Rules

Exponent rules are the secret sauce to simplifying and manipulating exponential expressions. They provide a set of guidelines that allow us to rewrite expressions in different forms without changing their value. Let's review some of the key rules that we'll use to solve our problem:

  1. Product of Powers Rule: When multiplying powers with the same base, we add the exponents. Mathematically, this looks like: b^m * b^n = b^(m+n).

  2. Quotient of Powers Rule: When dividing powers with the same base, we subtract the exponents: b^m / b^n = b^(m-n).

  3. Power of a Power Rule: When raising a power to another power, we multiply the exponents: (bm)n = b^(m*n).

  4. Power of a Product Rule: When raising a product to a power, we distribute the exponent to each factor: (ab)^n = a^n * b^n.

  5. Power of a Quotient Rule: When raising a quotient to a power, we distribute the exponent to both the numerator and the denominator: (a/b)^n = a^n / b^n.

  6. Negative Exponent Rule: A negative exponent indicates a reciprocal: b^(-n) = 1/b^n.

  7. Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1: b^0 = 1.

These rules might seem a bit abstract at first, but they become much clearer when we apply them to actual problems. So, let's get back to our original question and see how these rules can help us find an equivalent expression for 2^(3x-4).

Tackling the Problem: Finding the Equivalent Expression

Okay, guys, let's get down to business. Our mission is to find an expression that is equivalent to 2^(3x-4). This means we need to manipulate the expression using the exponent rules we just discussed until we arrive at one of the answer choices provided. The answer choices we have are:

A. 1^x/8

B. 3^x/4

C. 6^x/8

D. 8^x/16

Let's start by focusing on the exponent, 3x - 4. We can think of this exponent as a combination of multiplication and subtraction. This hints that we might be able to use the product and quotient of powers rules to rewrite the expression.

Step 1: Breaking Down the Exponent

The first thing we can do is use the quotient of powers rule in reverse. Remember, b^(m-n) = b^m / b^n. So, we can rewrite 2^(3x-4) as 2^(3x) / 2^4.

This step is crucial because it separates the variable term (3x) from the constant term (-4) in the exponent. Now we can deal with each part more easily.

Step 2: Simplifying the Constant Term

Let's simplify the denominator first. We know that 2^4 means 2 * 2 * 2 * 2, which equals 16. So, our expression now looks like 2^(3x) / 16.

Step 3: Manipulating the Variable Term

Now, let's focus on the numerator, 2^(3x). This looks like a power raised to another power, which means we can use the power of a power rule: (bm)n = b^(m*n). In this case, we can rewrite 2^(3x) as (23)x.

Why did we do this? Because now we have a simpler base (2^3) raised to the power of x. We know that 2^3 is 2 * 2 * 2, which equals 8. So, our expression becomes 8^x / 16.

Step 4: Comparing with Answer Choices

Guess what? We've arrived at one of the answer choices! Our simplified expression, 8^x / 16, matches option D. So, the equivalent expression for 2^(3x-4) is 8^x / 16.

Why Other Options are Incorrect

Now that we've found the correct answer, let's briefly discuss why the other options are incorrect. This will help solidify our understanding of exponential expressions and the rules we used.

A. 1^x / 8: This option is incorrect because 1 raised to any power is always 1. So, 1^x is always 1, and the expression becomes 1/8, which is not equivalent to 2^(3x-4).

B. 3^x / 4: This option has a different base (3) in the numerator, which cannot be obtained by manipulating the original expression using exponent rules.

C. 6^x / 8: Similar to option B, this option has a base (6) that cannot be derived from the original expression through exponent manipulation.

By understanding why the incorrect options are wrong, we reinforce our understanding of the correct approach and the underlying principles.

Key Takeaways and Tips for Success

We've successfully found the equivalent expression for 2^(3x-4) by applying the rules of exponents. Let's recap the key takeaways and some tips to help you ace similar problems in the future:

  • Master the Exponent Rules: Knowing the exponent rules inside and out is crucial for simplifying and manipulating exponential expressions. Practice applying them in different scenarios to build your fluency.

  • Break Down Complex Exponents: When dealing with complex exponents like 3x - 4, try to break them down into simpler parts using the quotient and product of powers rules.

  • Look for Opportunities to Apply the Power of a Power Rule: This rule is often the key to rewriting expressions in a form that matches the answer choices.

  • Simplify Step-by-Step: Don't try to do everything at once. Simplify the expression step-by-step, focusing on one rule or operation at a time. This will help you avoid errors and keep your work organized.

  • Practice, Practice, Practice: The more you practice, the more comfortable you'll become with exponential expressions and the rules that govern them. Work through various examples and challenge yourself with more complex problems.

Real-World Applications of Exponential Expressions

Exponential expressions aren't just abstract mathematical concepts; they have numerous applications in the real world. Here are a few examples:

  • Compound Interest: The formula for compound interest involves exponential expressions, allowing us to calculate how investments grow over time.

  • Population Growth: Exponential growth models are used to describe how populations of organisms increase over time.

  • Radioactive Decay: Radioactive decay follows an exponential decay model, which helps us determine the age of ancient artifacts through carbon dating.

  • Computer Science: Exponential functions are used in algorithms and data structures to analyze their efficiency and performance.

Understanding exponential expressions and their applications can open doors to many exciting fields and opportunities. So, keep practicing and exploring the fascinating world of mathematics!

Conclusion

We've successfully navigated the world of exponential expressions and found that 8^x / 16 is equivalent to 2^(3x-4). By understanding the fundamental rules of exponents and practicing step-by-step simplification, you can confidently tackle similar problems. Remember, math is a journey, not a destination. So, keep learning, keep exploring, and keep having fun!

If you have any questions or want to delve deeper into exponential expressions, feel free to ask. Happy math-solving, guys!