Hey guys! Let's break down this math problem together and simplify it step by step. We've got a cool expression here involving variables and exponents, and our mission is to make it look as clean and straightforward as possible, using only positive exponents. So, grab your thinking caps, and let's dive right in!
The Expression
Our expression is: $2 v^{-8} v^3 \cdot 7 y^{-4} u^{-1} \cdot 6 u y^{-9}$
Looks a bit complex, right? But don't worry, we're going to tackle it piece by piece. The main goal here is to simplify this expression by combining like terms and making sure all exponents are positive. This involves a few key rules of exponents that we'll use along the way. For instance, remember that when you multiply terms with the same base, you add their exponents. Also, a negative exponent means we need to take the reciprocal of the base raised to the positive exponent. Keeping these rules in mind, we'll transform this expression into its simplest form. Think of it as decluttering your room, but instead of clothes and books, we're dealing with variables and exponents! It’s all about making things neat and organized. So, let's roll up our sleeves and get started, turning this mathematical puzzle into something much easier to handle.
Step-by-Step Simplification
1. Multiply the Coefficients
First up, let's multiply the coefficients (the numbers in front of the variables). We have 2, 7, and 6. Multiplying these together is pretty straightforward:
$2 \cdot 7 \cdot 6 = 84$
So, our expression now starts with 84. This step is like setting the foundation for our simplified expression. By dealing with the numbers first, we clear the way to focus on the variables and their exponents. It's a bit like sorting your tools before starting a project; getting the numbers sorted makes the rest of the process smoother and more organized. We're essentially gathering all the numerical parts of our expression into one neat package, which will make the subsequent steps much easier to manage. Think of it as the first big piece of the puzzle falling into place, making the bigger picture clearer. Alright, with the coefficients handled, let's move on to the next part – the variables!
2. Combine 'v' Terms
Next, we'll handle the terms with the variable 'v'. We have $v^{-8}$ and $v^3$. When multiplying terms with the same base, we add the exponents:
$v^{-8} \cdot v^3 = v^{-8+3} = v^{-5}$
So, the 'v' terms combine to give us $v^{-5}$. But remember, our goal is to have only positive exponents in our final answer. This means we're not quite done with 'v' yet! The negative exponent tells us that we need to take the reciprocal of $v^5$. Think of it as a little mathematical hint that there’s one more step to make this term fit our positive exponent requirement. This step is crucial in ensuring our final answer adheres to the rules of the problem. It's like making sure all the pieces of a Lego set are facing the right way before you snap them together. So, we've identified the issue – the negative exponent – and we know the solution. We're on the right track to simplifying this expression completely.
3. Combine 'y' Terms
Now, let's tackle the 'y' terms. We have $y^{-4}$ and $y^{-9}$. Again, we add the exponents:
$y^{-4} \cdot y^{-9} = y^{-4 + (-9)} = y^{-13}$
So, the 'y' terms combine to $y^{-13}$. Just like with the 'v' term, we have a negative exponent here, which means we're not quite finished with 'y' yet. This negative exponent is another clue that we need to do a bit more work to get this term into its simplest form with a positive exponent. It’s like encountering a speed bump on the road – it slows us down momentarily, but we know how to navigate it. The negative exponent is simply telling us that this 'y' term needs to be moved to the denominator to become positive. We're systematically working through each variable, ensuring that every term meets our criteria for simplification. Think of it as decluttering each shelf in a cabinet, one at a time, until everything is perfectly organized.
4. Combine 'u' Terms
Time for the 'u' terms. We have $u^{-1}$ and $u$. Remember, $u$ is the same as $u^1$. So, we add the exponents:
$u^{-1} \cdot u^1 = u^{-1 + 1} = u^0$
Anything raised to the power of 0 is 1. So, $u^0 = 1$. This is a cool little twist! The 'u' terms have completely simplified to 1, meaning they won't even appear in our final simplified expression. It’s like finding a hidden shortcut in a maze – we’ve bypassed a potentially tricky part and made our journey to the solution even smoother. This outcome simplifies our expression significantly, as we can now effectively ignore the 'u' terms. It’s always satisfying when terms cancel out or simplify to such a fundamental value. This step highlights the elegance of mathematical simplification, where seemingly complex expressions can sometimes reduce to much simpler forms. We're making great progress, guys!
5. Rewrite with Positive Exponents
Now, let's put it all together. We have:
$84 \cdot v^{-5} \cdot y^{-13} \cdot 1 = 84v^{-5}y^{-13}$
Remember, we want only positive exponents. To make the exponents positive, we move the terms with negative exponents to the denominator:
$84v^{-5}y^{-13} = \frac{84}{v^5y^{13}}$
And there we have it! We've successfully rewritten the expression with positive exponents. This final step is like putting the finishing touches on a masterpiece. We've taken a complex expression and transformed it into a neat, simplified form. By moving the terms with negative exponents to the denominator, we've adhered to the rules of positive exponents, completing our mission. This transformation demonstrates the power of algebraic manipulation, where we can rearrange and simplify expressions to make them easier to understand and work with. It's like translating a sentence into a clearer, more concise form. The end result is a clean, elegant expression that showcases the beauty of mathematical order.
Final Answer
The simplified expression with only positive exponents is:
$\frac{84}{v^5y^{13}}$
Awesome job, guys! We took a complex expression and simplified it beautifully. Remember, the key is to break it down step by step and apply the rules of exponents. You've got this!