Simplify Exponential Expressions: A Step-by-Step Guide

Hey guys! Let's break down how to simplify the expression $10 m^7 n imes 7 m^4 n$ and express the answer using positive exponents. It might seem intimidating at first, but don't worry—we'll go through it together step by step. Understanding these kinds of problems is super important in algebra, and it’s actually pretty fun once you get the hang of it! So, let's dive in and make sure we nail this concept.

Understanding the Basics of Exponential Expressions

Before we jump into the problem, let’s quickly recap what exponential expressions are all about. In simple terms, an exponential expression consists of a base and an exponent. The base is the number or variable being multiplied, and the exponent tells us how many times the base is multiplied by itself. For example, in $m^7$, m is the base, and 7 is the exponent. This means we're multiplying m by itself seven times. Similarly, in $m^4$, m is multiplied by itself four times.

When we're dealing with multiple terms in an expression, like in our problem, we need to remember a few key rules. One of the most crucial rules is the product of powers rule. This rule states that when you multiply terms with the same base, you add the exponents. Mathematically, it looks like this: $a^m imes a^n = a^{m+n}$. This rule is a cornerstone for simplifying expressions involving exponents, and you'll see how we use it in our problem.

Another important point to remember is how to handle coefficients. Coefficients are the numbers that multiply the variables (like 10 and 7 in our expression). When simplifying, we treat these coefficients separately and multiply them as usual. So, it's like having two separate tasks: dealing with the numbers and dealing with the variables with exponents. This separation makes the whole process much clearer and less overwhelming. Now that we’ve covered the basics, let’s move on to tackling the actual problem.

Breaking Down the Expression $10 m^7 n imes 7 m^4 n$

Okay, let's get started! Our mission is to simplify the expression $10 m^7 n imes 7 m^4 n$. The first thing we want to do is identify the different parts of the expression: the coefficients, the variables, and their exponents. We have coefficients 10 and 7, and variables m and n with their respective exponents. Breaking it down like this helps us organize our thoughts and approach the problem systematically.

Next, we'll rearrange the expression to group like terms together. This means putting the coefficients next to each other and the variables with the same base next to each other. So, we rewrite the expression as $10 imes 7 imes m^7 imes m^4 imes n imes n$. Grouping like terms makes it easier to see what operations we need to perform. It's all about making the complex look simple!

Now, let's focus on the coefficients. We simply multiply 10 by 7, which gives us 70. So, we've taken care of the numerical part of the expression. Next up are the variables with exponents. This is where the product of powers rule comes into play. Remember, when multiplying terms with the same base, we add the exponents. This is the golden rule that will help us simplify the variable part of our expression. So, let's see how we can apply this to the m and n terms.

Applying the Product of Powers Rule

Time to put our rule into action! We have $m^7 imes m^4$. Both terms have the same base (m), so we can apply the product of powers rule. We add the exponents 7 and 4, which gives us $7 + 4 = 11$. Therefore, $m^7 imes m^4$ simplifies to $m^{11}$. See? It’s not as scary as it looks!

Now, let's tackle the n terms. We have $n imes n$. When a variable doesn't have an explicit exponent, it's understood to have an exponent of 1. So, we can rewrite $n$ as $n^1$. This means we have $n^1 imes n^1$. Again, we apply the product of powers rule and add the exponents: $1 + 1 = 2$. Thus, $n imes n$ simplifies to $n^2$. We've now handled all the variable parts of our expression.

So far, we've simplified the coefficients to 70, $m^7 imes m^4$ to $m^{11}$, and $n imes n$ to $n^2$. Now it’s just a matter of putting it all together. We’re almost there, guys! The key to success in these kinds of problems is breaking them down into manageable steps and applying the rules we know. Let’s wrap this up and get to our final simplified expression.

Putting It All Together for the Final Answer

Alright, we've done the heavy lifting! We know that $10 imes 7$ is 70, $m^7 imes m^4$ simplifies to $m^{11}$, and $n imes n$ simplifies to $n^2$. Now, we just need to combine these simplified parts to get our final answer. This is the moment of truth, so let's make sure we put everything in the right place.

To combine the terms, we simply write them next to each other. So, we have 70, $m^{11}$, and $n^2$. Putting it all together, our simplified expression is $70m^{11}n^2$. This is our final answer! We've successfully simplified the original expression and expressed it using positive exponents.

Remember, the key to simplifying these types of expressions is to break them down into smaller, manageable parts. First, we deal with the coefficients by multiplying them. Then, we focus on the variables with exponents, applying the product of powers rule where necessary. Finally, we combine the simplified parts to get our final answer. By following these steps, you can tackle even the most complex exponential expressions with confidence. Great job, guys! You’ve got this!

Practice Problems to Sharpen Your Skills

Now that we've walked through the solution, let's reinforce what we've learned with some practice problems. Solving more problems will help you solidify your understanding and make you more comfortable with simplifying exponential expressions. Practice makes perfect, as they say!

Here are a few problems for you to try:

1. Simplify: $5x^3y^2 imes 3x^2y^4$
2. Simplify: $8a^5b imes 2a^2b^3$
3. Simplify: $4p^6q^3 imes 6pq^5$

Work through these problems step by step, just like we did in the example. Remember to multiply the coefficients, apply the product of powers rule for variables with the same base, and combine your results. Don't rush; take your time and focus on applying the rules correctly. If you get stuck, review the steps we discussed earlier in the article.

Once you've solved these problems, check your answers. If you got them right, fantastic! You're well on your way to mastering exponential expressions. If you made any mistakes, don't worry. Go back and see where you went wrong, and try again. Learning from our mistakes is a crucial part of the learning process. Keep practicing, and you'll find that these types of problems become second nature. So, grab a pen and paper, and let’s get to work!

Common Mistakes to Avoid

As we wrap up, let's chat about some common pitfalls to watch out for when simplifying exponential expressions. Knowing these common mistakes can save you a lot of headaches and help you avoid errors in your work. It's like having a map of the danger zones so you can steer clear!

One frequent mistake is forgetting to add the exponents when multiplying terms with the same base. Remember, the product of powers rule is your friend here: $a^m imes a^n = a^{m+n}$. It's easy to accidentally multiply the exponents instead of adding them, so always double-check this step. Another slip-up is not accounting for coefficients. Don't forget to multiply the numerical coefficients separately from the variables. It’s like they're two different teams working together, so treat them individually before combining them.

Another common mistake occurs when dealing with variables that appear without an explicit exponent. As we discussed earlier, a variable without an exponent is understood to have an exponent of 1. Forgetting this can lead to incorrect simplifications, especially when applying the product of powers rule. Always remember that $n$ is the same as $n^1$.

Finally, be careful with the order of operations. Make sure you're applying the rules in the correct sequence. Usually, you'll want to handle coefficients first, then apply the product of powers rule to variables with the same base. Keeping these common mistakes in mind will help you tackle these problems with greater accuracy and confidence. So, stay sharp, guys, and keep those rules straight!

By understanding the basics, breaking down the expression, applying the product of powers rule, and avoiding common mistakes, you'll be simplifying exponential expressions like a pro in no time! Keep practicing, and you'll master these skills in no time!