Simplify: Positive Exponents Explained

Hey guys, let's dive into simplifying expressions with exponents! Today, we're tackling a problem that looks a bit intimidating at first glance, but trust me, it's all about breaking it down step-by-step. We're going to simplify the expression $\left(3 u v w^{-1}\right)\left(2 u^{-8} v w\right)\left(4 u^0 v^9 w^{-4}\right)$ and express our answer using positive exponents only. This is a super common task in algebra, and once you get the hang of the rules, you'll be simplifying like a pro in no time. So, grab your notebooks, maybe a snack, and let's get this done together!

Understanding the Basics: Exponent Rules are Your Best Friends

Before we jump into the main event, let's quickly recap some fundamental exponent rules that will be our guiding stars. These rules are the bedrock of simplifying any expression involving powers. First up, we have the Product of Powers Rule: when you multiply terms with the same base, you add their exponents. So, $x^a \cdot x^b = x^{a+b}$. Next, the Power of a Power Rule: when you raise a power to another power, you multiply the exponents. That is, $(x^a)^b = x^{ab}$. And crucially for this problem, the Zero Exponent Rule: any non-zero number raised to the power of zero is just 1. So, $x^0 = 1$. Finally, the Quotient of Powers Rule: when you divide terms with the same base, you subtract their exponents. $x^a / x^b = x^{a-b}$. And the Negative Exponent Rule: a term with a negative exponent in the numerator can be moved to the denominator with a positive exponent, and vice versa. $x^{-a} = 1/x^a$ and $1/x^{-a} = x^a$. Knowing these rules is key to conquering our problem. We'll be applying these throughout our simplification process, so keep them handy!

Step-by-Step Simplification: Let's Break It Down!

Alright, let's get down to business with our expression: \\left(3 u v w^{-1}\right)\\left(2 u^{-8} v w\right)\\left(4 u^0 v^9 w^{-4}\right). The first thing we want to do is group the coefficients (the numbers) and then group the variables with the same base together. This makes it much easier to manage. So, we have:

(3 \cdot 2 \cdot 4) \cdot (u \cdot u^{-8} \cdot u^0) \cdot (v \cdot v \cdot v^9) \cdot (w^{-1} \cdot w \cdot w^{-4})$

Let's tackle each group one by one.

Simplifying the Coefficients:

This is the easiest part, guys! We just multiply the numbers together: $3 \cdot 2 \cdot 4 = 6 \cdot 4 = 24$. Easy peasy, right?

Simplifying the 'u' terms:

Now for the 'u' variables. Remember the Product of Powers Rule ($x^a \cdot x^b = x^{a+b}$)? We'll apply it here. The exponents for 'u' are 1 (from $u$, which is $u^1$), -8, and 0 (from $u^0$). So, we add them up: $u^1 \cdot u^{-8} \cdot u^0 = u^{1 + (-8) + 0} = u^{1 - 8 + 0} = u^{-7}$. We've got a negative exponent here, which we'll deal with later to ensure our final answer has only positive exponents.

Simplifying the 'v' terms:

Moving on to the 'v' variables. Again, we use the Product of Powers Rule. The exponents for 'v' are 1 (from $v$, which is $v^1$), 1 (from $v$), and 9. Adding them up, we get: $v^1 \cdot v^1 \cdot v^9 = v^{1 + 1 + 9} = v^{11}$. This one is already a positive exponent, which is great!

Simplifying the 'w' terms:

Finally, let's simplify the 'w' variables. The exponents are -1, 1 (from $w$, which is $w^1$), and -4. Applying the Product of Powers Rule: $w^{-1} \cdot w^1 \cdot w^{-4} = w^{-1 + 1 + (-4)} = w^{-1 + 1 - 4} = w^{-4}$. Another negative exponent to sort out.

Assembling the Simplified Expression

Now, let's put all our simplified parts back together. We have:

$24 \cdot u^{-7} \cdot v^{11} \cdot w^{-4}$

This is our expression after applying the product rule to each variable. However, the requirement is to express the answer using positive exponents only. So, we need to address those negative exponents using the Negative Exponent Rule ($x^{-a} = 1/x^a$).

• The $u^{-7}$ term needs to move to the denominator and become $u^7$. So, it becomes $1/u^7$.
• The $w^{-4}$ term also needs to move to the denominator and become $w^4$. So, it becomes $1/w^4$.

The $v^{11}$ term already has a positive exponent, so it stays in the numerator.

So, our expression transforms into:

$24 \cdot \frac{1}{u^7} \cdot v^{11} \cdot \frac{1}{w^4}$

To write this as a single fraction, we multiply everything in the numerator and everything in the denominator:

$\frac{24 \cdot v^{11}}{u^7 \cdot w^4}$

Final Check and Conclusion

And there you have it, guys! The simplified expression with only positive exponents is $\frac{24 v^{11}}{u^7 w^4}$. We successfully navigated through combining terms, applying exponent rules like the product of powers and zero exponent rule, and finally converted negative exponents into positive ones by moving them across the fraction bar. Remember, the key is to stay organized, identify the bases, and apply the correct rule for each step. It might seem like a lot at first, but with practice, these kinds of problems become second nature. Keep practicing, and you'll master exponents in no time! If you ever get stuck, just go back to the basic rules – they're your best friends in the world of algebra. Keep up the great work, mathletes!