Simplifying Exponential Expressions: A Step-by-Step Guide

Hey guys! Ever get tangled up in those pesky exponential expressions? Don't worry, we've all been there. Today, we're going to break down how to simplify expressions like $2 w^9 y^5 \cdot 3 y^7 \cdot 7 w$ step by step. By the end of this guide, you'll be a pro at simplifying these types of problems.

Understanding the Basics

Before we dive into the problem, let's quickly recap the basic rules of exponents. Understanding these rules is crucial for simplifying any exponential expression. So, pay close attention, alright?

What are Exponents?

At its core, an exponent tells you how many times to multiply a base by itself. For example, in the term $x^3$, 'x' is the base, and '3' is the exponent. This means you multiply 'x' by itself three times: $x * x * x$. Simple enough, right? This foundational understanding is key as we move forward.

Key Rules of Exponents

Here are a few rules we'll be using today:

1. Product of Powers Rule: When multiplying terms with the same base, you add the exponents. Mathematically, it's expressed as $a^m * a^n = a^{m+n}$. Think of it like this: if you have $x^2 * x^3$, you're really multiplying (x * x) by (x * x * x), which gives you $x^5$. This is a cornerstone rule.
2. Commutative Property: Remember, multiplication is commutative, meaning the order doesn't matter. So, $a * b = b * a$. This allows us to rearrange terms to group like bases together. This flexibility is super handy.
3. Associative Property: Multiplication is also associative, which means how we group the terms doesn't change the result. So, $(a * b) * c = a * (b * c)$. This lets us multiply coefficients separately from the variables. Grouping makes things easier!

Now that we've got the basics down, let's tackle our problem.

Step 1: Identify and Group Like Terms

Our expression is $2 w^9 y^5 \cdot 3 y^7 \cdot 7 w$. The first thing we want to do is identify the like terms. By like terms, we mean terms that have the same variable base. In our expression, we have terms with 'w' and terms with 'y'.

Let's rewrite the expression, grouping the like terms together. Thanks to the commutative property, we can rearrange the terms without changing the result:

$2 w^9 y^5 \cdot 3 y^7 \cdot 7 w = 2 \cdot 3 \cdot 7 \cdot w^9 \cdot w \cdot y^5 \cdot y^7$

See how we've grouped the coefficients (2, 3, and 7), the 'w' terms ($w^9$ and $w$), and the 'y' terms ($y^5$ and $y^7$)? This makes the next steps much clearer. This rearrangement is all about organization.

Step 2: Multiply the Coefficients

Next, let's multiply the coefficients together. Coefficients are the numerical parts of the terms. In our grouped expression, the coefficients are 2, 3, and 7. Multiplying these together is straightforward:

$2 \cdot 3 \cdot 7 = 6 \cdot 7 = 42$

So, the numerical part of our simplified expression will be 42. We've handled the numerical part!

Step 3: Simplify the 'w' Terms

Now, let's focus on the 'w' terms: $w^9 \cdot w$. Remember, when a variable doesn't have an explicitly written exponent, it's understood to have an exponent of 1. So, 'w' is the same as $w^1$.

Using the product of powers rule, which states that $a^m * a^n = a^{m+n}$, we can simplify this:

$w^9 \cdot w^1 = w^{9+1} = w^{10}$

So, the 'w' part of our simplified expression is $w^{10}$. We're making progress!

Step 4: Simplify the 'y' Terms

Now, let's tackle the 'y' terms: $y^5 \cdot y^7$. Again, we'll use the product of powers rule:

$y^5 \cdot y^7 = y^{5+7} = y^{12}$

So, the 'y' part of our simplified expression is $y^{12}$. Almost there!

Step 5: Combine Everything

We've simplified the coefficients, the 'w' terms, and the 'y' terms. Now, all that's left is to put everything together. We have:

• Coefficients: 42
• 'w' terms: $w^{10}$
• 'y' terms: $y^{12}$

Combining these gives us our final simplified expression:

$42w^{10}y^{12}$

And that's it! We've successfully simplified the expression. Victory!

Common Mistakes to Avoid

Simplifying exponential expressions can be tricky, and it's easy to make mistakes if you're not careful. Here are a few common pitfalls to watch out for:

• Forgetting the Exponent of 1: As we mentioned earlier, if a variable doesn't have an explicitly written exponent, it's understood to be 1. Forgetting this can lead to errors when applying the product of powers rule. Always remember the implicit 1!
• Adding Coefficients When You Shouldn't: The product of powers rule applies only to terms with the same base. You should never add the coefficients when multiplying terms with exponents. Coefficients stay separate!
• Misapplying the Product of Powers Rule: Make sure you're adding exponents only when multiplying terms with the same base. Don't try to apply this rule to terms with different bases or to coefficients. Rule application matters!
• Skipping Steps: It's tempting to rush through the simplification process, but skipping steps can increase the likelihood of making mistakes. Take your time, and write out each step clearly. Patience is key!

Practice Makes Perfect

The best way to master simplifying exponential expressions is to practice! Try working through similar problems on your own. The more you practice, the more comfortable you'll become with the rules and techniques. Practice, practice, practice!

Here's a quick recap of the steps we followed:

1. Identify and Group Like Terms: This sets the stage for simplification.
2. Multiply the Coefficients: Deal with the numerical part first.
3. Simplify Variable Terms: Use the product of powers rule.
4. Combine Everything: Put the simplified parts together.

Conclusion

Simplifying exponential expressions might seem daunting at first, but by breaking it down into manageable steps and understanding the basic rules, it becomes much easier. Remember to group like terms, apply the product of powers rule correctly, and avoid common mistakes. Keep practicing, and you'll be simplifying like a pro in no time! You got this! And if you have any questions, don't hesitate to ask. Happy simplifying!