Hey everyone! Today, we're diving into a fun math problem that involves simplifying an expression with exponents. Our main goal? To break it down into something super manageable, showing only two powers. Let's jump right into it!
Understanding the Problem
First off, let's make sure we all understand the problem we're tackling. We've got this expression: 5⁵(16² × 5³)³. It looks a bit intimidating, right? But don't worry, we're going to break it down step by step. Our mission is to simplify this so it only shows two powers – that means we want to end up with something like *aˣ * bʸ, where a and b are different numbers (our 'bases') and x and y are their exponents. So, in essence, we need to simplify this complicated expression into a more digestible format, focusing on reducing it to its simplest form, expressing it with minimal components.
To get there, we'll need to remember a few key rules about exponents. Think of exponents as a shorthand way of showing how many times a number is multiplied by itself. For example, 5⁵ means 5 * 5 * 5 * 5 * 5. Understanding this basic concept is crucial. Then, we'll use rules like the power of a power rule (where (aˣ)ʸ = aˣ*ʸ) and the product of powers rule (where aˣ * aʸ = aˣ+ʸ). These rules are the tools we'll use to simplify the expression. They help us manipulate the exponents and bases to combine like terms and reduce the complexity of the problem. We’ll also need to recognize that 16 is a power of 2 (16 = 2⁴), which will help us combine terms later on. This is a common strategy in simplifying expressions: look for ways to rewrite numbers as powers of a common base. This makes it much easier to combine terms and simplify the overall expression. So, let's gear up and get ready to simplify!
Breaking Down the Expression
Alright, let's start by breaking down the expression 5⁵(16² × 5³)³ step-by-step. The first thing we need to tackle is the part inside the parentheses: (16² × 5³). Remember, we want to simplify this, and a key step is recognizing that 16 is a power of 2. Specifically, 16 equals 2⁴. This is a crucial observation because it allows us to rewrite the expression in terms of prime factors, which will make our calculations much easier. So, let’s rewrite 16² as (2⁴)². Now, using the power of a power rule, which states that (aˣ)ʸ = aˣ*ʸ, we can simplify (2⁴)² to 2⁸. This is a significant step because we've reduced 16², which initially seemed complex, to a simple power of 2. This kind of transformation is what simplifying expressions is all about – making complex things simple.
Now our expression inside the parentheses looks like this: (2⁸ × 5³). Much cleaner, right? But we're not done yet! We still have that exponent of 3 outside the parentheses to deal with. This means we need to raise the entire expression (2⁸ × 5³) to the power of 3. To do this, we again use the power of a power rule. This time, we're applying it to a product: (2⁸ × 5³ )³. The rule tells us that we need to raise each factor inside the parentheses to the power of 3. So, (2⁸)³ becomes 2^(83) = 2²⁴, and (5³)³ becomes 5^(33) = 5⁹. Remember, guys, each component inside the parenthesis will be powered by 3. Now, our original expression 5⁵(16² × 5³)³ has transformed into 5⁵(2²⁴ × 5⁹). See how far we've come? We've successfully tackled the parentheses and the outer exponent, leaving us with a much simpler expression. Next, we’ll combine the powers of 5 to get our final simplified form.
Combining Powers
Okay, so we've simplified our expression to 5⁵(2²⁴ × 5⁹). Now comes the really cool part: combining the powers. We've got two terms with the same base, 5: one with an exponent of 5 (5⁵) and another with an exponent of 9 (5⁹). When you're multiplying terms with the same base, you can use the product of powers rule, which says aˣ * aʸ = aˣ+ʸ. This rule is super handy for simplifying expressions like ours.
So, let's apply this rule to our expression. We have 5⁵ multiplied by 5⁹. According to the rule, we simply add the exponents: 5 + 9 = 14. This means 5⁵ * 5⁹ simplifies to 5¹⁴. Awesome, right? We've taken two separate terms and combined them into one. Now our expression looks even cleaner: 2²⁴ × 5¹⁴. Notice that we're keeping the term with the base 2 (2²⁴) as it is because there are no other terms with the same base to combine it with. It's like having different ingredients in a recipe – you only mix the ones that are the same!
At this stage, we've done the bulk of the work. We've simplified the expression by dealing with the parentheses, applying the power of a power rule, and combining like terms using the product of powers rule. We're left with 2²⁴ × 5¹⁴, which is exactly what we wanted: an expression with only two powers. We've simplified a complex-looking expression into something much more manageable. This shows the power of understanding exponent rules and applying them strategically. In the next section, we'll wrap things up and highlight the key steps we took to achieve this simplification.
Final Simplified Form
Alright, guys, let's bring it all together! We started with the expression 5⁵(16² × 5³)³, and after a series of simplifications, we've arrived at the final form: 2²⁴ × 5¹⁴. How cool is that? We've successfully reduced a seemingly complex expression into something much simpler, showing only two powers.
To recap, here's what we did: First, we recognized that 16 is a power of 2 (16 = 2⁴), which allowed us to rewrite the expression in terms of prime factors. This is a crucial step because it sets the stage for applying exponent rules more effectively. Then, we used the power of a power rule to simplify (2⁴)² to 2⁸. This rule is like a secret weapon for dealing with exponents raised to other exponents – it allows you to multiply the exponents together and simplify the expression. Next, we dealt with the exponent outside the parentheses by applying the power of a power rule again. We raised each factor inside the parentheses to the power of 3, which gave us 2²⁴ × 5⁹. This step is where things really started to come together, as we were able to distribute the outer exponent and simplify each term individually.
Finally, we combined the powers of 5 using the product of powers rule. This rule is another key tool in our exponent-simplifying arsenal. By adding the exponents of terms with the same base, we simplified 5⁵ * 5⁹ to 5¹⁴. This left us with our final, simplified form: 2²⁴ × 5¹⁴. This final step demonstrates the elegance of exponent rules – they allow us to combine terms and simplify expressions in a very systematic way.
Key Takeaways
So, what are the key takeaways from this simplification journey? First and foremost, understanding exponent rules is crucial. The power of a power rule (aˣ)ʸ = aˣ*ʸ and the product of powers rule aˣ * aʸ = aˣ+ʸ are your best friends when it comes to simplifying expressions with exponents. These rules are like the fundamental building blocks of exponent manipulation, and mastering them will make your life much easier. Another important thing is to look for opportunities to rewrite numbers as powers of a common base. Recognizing that 16 = 2⁴ was a game-changer in this problem, as it allowed us to combine terms more easily. This is a common strategy in simplifying expressions, and it's something you should always be on the lookout for. Lastly, remember to break down complex problems into smaller, manageable steps. We tackled this problem piece by piece, dealing with the parentheses first, then the outer exponent, and finally combining like terms. This step-by-step approach makes the problem much less daunting and allows you to focus on each step individually.
In conclusion, by mastering exponent rules, recognizing opportunities to rewrite numbers, and breaking down complex problems, you can simplify even the most intimidating expressions. Keep practicing, and you'll become an exponent simplification pro in no time!