Simplifying Exponential Expressions: A Math Problem Solved
Hey guys! Today, we're diving into a fascinating math problem that involves simplifying exponential expressions. This is a common topic in algebra, and mastering it can really boost your problem-solving skills. We'll break down the problem step-by-step, so you can follow along easily. Let's jump right into it!
Understanding the Problem
Okay, so here’s the problem we’re tackling: What is the simplest form of the expression: (9³•9⁴-27²•3⁴)/(3¹⁰ -9⁴)? This looks a bit intimidating at first glance, but don't worry! We're going to take it apart piece by piece. The key to solving this is understanding the rules of exponents and how to manipulate them. Remember, exponential expressions are just a shorthand way of writing repeated multiplication. For example, 9³ means 9 multiplied by itself three times (9 * 9 * 9).
When you first look at a complex expression like this, it’s super important to identify the different components. We've got numbers raised to various powers, multiplication, subtraction, and division all happening in one place. The order of operations (PEMDAS/BODMAS) will be our guiding light here. We’ll deal with the exponents first, then multiplication, followed by subtraction, and finally division. But before we start crunching numbers, let's see if we can simplify the expression by expressing all the terms with the same base. This is a common strategy when dealing with exponents, and it can make the calculations much easier. Think of it like trying to compare apples and oranges – it’s easier if you can convert everything into apples (or oranges!).
Remember, simplifying exponential expressions often involves rewriting numbers in terms of a common base. In this case, we can express 9 and 27 as powers of 3. This will allow us to combine terms more easily and eventually simplify the entire expression. So, keep this strategy in mind as we move forward. It’s all about breaking down the complex into manageable parts and applying the rules of exponents systematically. Now, let's get our hands dirty and start simplifying!
Breaking Down the Expression
Alright, let's start breaking down this expression. The first thing we want to do is express all the numbers in terms of the same base. Looking at the expression (9³•9⁴-27²•3⁴)/(3¹⁰ -9⁴), we can see that 9 and 27 can both be written as powers of 3. This is a crucial step because it allows us to use the rules of exponents more effectively. Remember, the goal here is to make the expression easier to handle. By having the same base, we can combine exponents using various rules, like the product of powers rule or the quotient of powers rule.
So, let's rewrite 9 and 27 as powers of 3. We know that 9 is 3², and 27 is 3³. Replacing these in the expression, we get: ((3²)³•(3²)⁴-(3³)²•3⁴)/(3¹⁰ -(3²)⁴). Now, this looks a little more manageable, right? We've successfully expressed all the terms with a base of 3. The next step is to apply the power of a power rule, which states that (am)n = a^(m*n). This rule is super handy when we have an exponent raised to another exponent. It allows us to simplify these terms by multiplying the exponents.
Applying the power of a power rule, we can simplify the expression further. (3²)³ becomes 3^(23) = 3⁶, (3²)⁴ becomes 3^(24) = 3⁸, (3³)² becomes 3^(32) = 3⁶, and (3²)⁴ becomes 3^(24) = 3⁸. So, our expression now looks like this: (3⁶•3⁸-3⁶•3⁴)/(3¹⁰ -3⁸). See how much simpler it's becoming? By using the power of a power rule, we've reduced the complexity of the exponents. Now, we can move on to the next step, which involves using the product of powers rule to combine terms in the numerator. We're making good progress, guys! Keep following along, and you'll see how all these steps come together to simplify the expression.
Applying Exponent Rules
Okay, so we've rewritten our expression as (3⁶•3⁸-3⁶•3⁴)/(3¹⁰ -3⁸). Now, it's time to apply some more exponent rules to further simplify things. The first rule we'll use is the product of powers rule, which states that a^m • a^n = a^(m+n). This rule is essential for combining terms with the same base that are being multiplied. In our expression, we have 3⁶ multiplied by 3⁸ and 3⁶ multiplied by 3⁴ in the numerator. By applying the product of powers rule, we can simplify these terms.
Let's start with 3⁶•3⁸. According to the product of powers rule, this is equal to 3^(6+8) = 3¹⁴. Similarly, 3⁶•3⁴ becomes 3^(6+4) = 3¹⁰. So, our expression now looks like this: (3¹⁴-3¹⁰)/(3¹⁰ -3⁸). See how we're gradually reducing the complexity? By using the product of powers rule, we've combined the exponents in the numerator, making it easier to see the next steps. The expression is definitely starting to look less intimidating!
The next thing we want to do is factor out the common term in the numerator. This is a crucial step in simplifying expressions, as it allows us to cancel out terms later on. Looking at the numerator (3¹⁴-3¹⁰), we can see that both terms have 3¹⁰ as a common factor. Factoring out 3¹⁰, we get 3¹⁰(3⁴-1). This makes the expression even more manageable. Now, our expression is 3¹⁰(3⁴-1)/(3¹⁰ -3⁸). We're making great progress, guys! Factoring out the common term was a key step, and now we can move on to simplifying the denominator and see if we can cancel out any terms. Keep up the good work!
Factoring and Simplifying
Alright, we've got our expression as 3¹⁰(3⁴-1)/(3¹⁰ -3⁸). The next step is to simplify the denominator. Looking at the denominator (3¹⁰ -3⁸), we can see that both terms have a common factor of 3⁸. Just like we did in the numerator, we're going to factor out this common term. Factoring is a powerful technique in simplifying expressions, and it often reveals hidden opportunities for cancellation. Remember, the goal is to make the expression as simple as possible, and factoring is a key tool in our arsenal.
Factoring out 3⁸ from the denominator, we get 3⁸(3²-1). So, our expression now looks like this: 3¹⁰(3⁴-1)/3⁸(3²-1). We're getting closer to the final simplified form! By factoring the denominator, we've created a situation where we can potentially cancel out terms. This is always a satisfying step because it means we're making significant progress towards the solution.
Now that we've factored both the numerator and the denominator, we can apply the quotient of powers rule to simplify the expression further. The quotient of powers rule states that a^m / a^n = a^(m-n). In our case, we have 3¹⁰ in the numerator and 3⁸ in the denominator. Applying the rule, 3¹⁰/3⁸ becomes 3^(10-8) = 3². So, our expression simplifies to 3²(3⁴-1)/(3²-1). We're on the home stretch now, guys! Canceling out the common factors using the quotient of powers rule has brought us much closer to the final answer. All that's left is to evaluate the remaining terms and simplify the expression completely.
Final Simplification and Solution
Okay, we've reached the final stage of simplifying our expression! We're currently at 3²(3⁴-1)/(3²-1). Now, it's time to evaluate the remaining terms and see if we can simplify further. This involves calculating the values of the exponents and then performing any remaining operations. Remember, we're aiming for the simplest form of the expression, so we need to make sure everything is fully simplified.
Let's start by evaluating the exponents. 3² is 3 * 3 = 9, 3⁴ is 3 * 3 * 3 * 3 = 81. So, our expression becomes 9(81-1)/(9-1). Now, we can perform the subtractions within the parentheses. 81 - 1 = 80, and 9 - 1 = 8. So, the expression is now 9(80)/8. We're almost there! We've reduced the expression to a simple arithmetic problem.
Next, we can perform the multiplication and division. 9 multiplied by 80 is 720. So, our expression is 720/8. Finally, dividing 720 by 8, we get 90. So, the simplest form of the expression is 90. Yay! We did it!
So, to recap, the simplest form of the expression (9³•9⁴-27²•3⁴)/(3¹⁰ -9⁴) is 90. We started with a complex expression and, by systematically applying the rules of exponents and factoring, we were able to simplify it to a single number. This is a great example of how breaking down a problem into smaller steps can make even the most challenging math problems manageable. Remember, practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time!
Conclusion
Guys, we've successfully simplified a complex exponential expression! Remember, the key to tackling these types of problems is to break them down into manageable steps. We started by rewriting the numbers with a common base, then applied the power of a power rule, the product of powers rule, and the quotient of powers rule. Factoring out common terms was also a crucial step in simplifying the expression. Finally, we evaluated the remaining terms to arrive at the simplest form.
This exercise demonstrates the importance of understanding and applying the rules of exponents. These rules are fundamental in algebra and are used in many different areas of mathematics. By mastering them, you'll be well-equipped to solve a wide range of problems. Keep practicing, and don't be afraid to tackle challenging expressions. With a systematic approach and a good understanding of the rules, you can simplify anything!
I hope this breakdown was helpful for you guys. Remember, math can be fun and rewarding when you approach it step by step. Keep exploring and keep learning!