Simplifying Exponential Expressions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the exciting world of simplifying exponential expressions. Specifically, we'll be tackling the problem: $\frac{2 p q^{-1}}{p^7 q}$. Our goal? To rewrite this expression using only positive exponents. Don't worry if this sounds a bit intimidating; we'll break it down into easy-to-understand steps, making sure you grasp the concepts. So, grab your pencils, and let's get started!

Understanding the Basics of Exponents

Before we jump into the simplification, let's refresh our memory on some fundamental rules of exponents. These rules are the key to unlocking the problem, so paying attention here is super important. First off, what exactly is an exponent? An exponent tells us how many times a base number is multiplied by itself. For example, in the expression $x^3$, 'x' is the base, and '3' is the exponent, meaning $x$ is multiplied by itself three times ($x * x * x$).

Now, let's look at some crucial exponent rules:

• Product Rule: When multiplying terms with the same base, you add the exponents. For instance, $x^m * x^n = x^{m+n}$.
• Quotient Rule: When dividing terms with the same base, you subtract the exponents. So, $x^m / x^n = x^{m-n}$.
• Power of a Power Rule: When raising a power to another power, you multiply the exponents: $(x^m)^n = x^{m*n}$.
• Negative Exponent Rule: A term with a negative exponent can be rewritten as its reciprocal with a positive exponent: $x^{-n} = 1/x^n$. This is the rule we'll be using extensively in our problem.

Mastering these rules will turn you into an exponent superhero in no time. Let's start applying these rules to simplify our expression, which is $\frac{2 p q^{-1}}{p^7 q}$. Remember, practice makes perfect, so don't hesitate to work through many examples. The more you practice, the more confident you'll become! Let's get our hands dirty now and make those exponents sing!

Step-by-Step Simplification: Turning Negatives into Positives

Alright, let's roll up our sleeves and tackle the problem $\frac{2 p q^{-1}}{p^7 q}$. Our first objective is to get rid of that pesky negative exponent, $q^{-1}$. This is where the negative exponent rule comes into play. Recall that $x^{-n} = 1/x^n$. So, $q^{-1}$ is equivalent to $1/q$.

Let's rewrite our original expression, replacing $q^{-1}$ with $1/q$. This gives us:

$\frac{2 p (1/q)}{p^7 q}$

Now, simplify the numerator. We're essentially multiplying $2p$ by $1/q$, which results in:

$\frac{2 p}{q * p^7 q}$

Next, notice that we have $q$ multiplied by $q$ in the denominator. Remembering that $q$ has an implied exponent of 1 ($q^1$), we can apply the product rule, which states that when multiplying terms with the same base, you add the exponents. This gives us $q^1 * q^1 = q^{1+1} = q^2$.

Our expression now looks like this:

$\frac{2p}{p^7 q^2}$

We're making great progress! We've successfully eliminated the negative exponent and simplified the expression a bit. Now, we'll continue simplifying by addressing the 'p' terms. Stay with me, because the next steps are going to be key to complete our simplification. Keep practicing, and you'll find it gets easier every time you solve these equations!

Combining Like Terms: Dealing with 'p' in the Expression

We're now at the stage where we need to combine the 'p' terms in our expression $\frac{2p}{p^7 q^2}$. Notice that we have 'p' in the numerator (which is essentially $p^1$) and $p^7$ in the denominator. To simplify, we'll apply the quotient rule, which states that when dividing terms with the same base, you subtract the exponents. This means we'll subtract the exponent in the denominator from the exponent in the numerator.

So, $p^1 / p^7 = p^{1-7} = p^{-6}$.

Now our expression looks like this:

$\frac{2 p^{-6}}{q^2}$

But wait! Remember, our goal is to express everything with positive exponents. We still have that pesky negative exponent on 'p'. Time to use the negative exponent rule again: $p^{-6} = 1/p^6$. We can rewrite our expression:

$\frac{2 * (1/p^6)}{q^2}$

This simplifies to:

$\frac{2}{p^6 q^2}$

And there you have it! The expression $\frac{2 p q^{-1}}{p^7 q}$ has been successfully simplified to $\frac{2}{p^6 q^2}$, and we've done it using only positive exponents. Pretty cool, right? You should also remember that practice is really important when you're learning these kinds of things. Make sure you practice every day and try different equations. Let's move on and summarize our key takeaways.

Final Answer and Key Takeaways

Congratulations, we did it! We successfully simplified $\frac{2 p q^{-1}}{p^7 q}$ to $\frac{2}{p^6 q^2}$ using positive exponents. The key steps involved:

1. Addressing Negative Exponents: We used the rule $x^{-n} = 1/x^n$ to eliminate the negative exponent on 'q'.
2. Combining Terms: We simplified the 'q' terms in the denominator using the product rule.
3. Applying the Quotient Rule: We used the quotient rule to simplify the 'p' terms.
4. Final Simplification: We applied the negative exponent rule one last time to ensure all exponents were positive.

Remember, mastering exponent rules is like having a superpower in algebra. Practice these rules regularly by working through different examples, and you'll become more confident in your ability to simplify exponential expressions. Always remember to break down the problems into small parts and tackle them one by one. This approach will make the whole process much easier.

Quick Recap of the Rules We Used:

• Product Rule: $x^m * x^n = x^{m+n}$
• Quotient Rule: $x^m / x^n = x^{m-n}$
• Negative Exponent Rule: $x^{-n} = 1/x^n$

Keep practicing, and you'll become an exponent wizard in no time! Always remember that math, like everything else, takes time and practice to master. Keep going, and you'll definitely reach your goals. Good luck on your math journey, guys! Feel free to revisit this guide and practice different problems, and you'll be an expert in no time!