Simplifying Expressions: (3a^2b^7)(5a^3b^8) Solution

Hey there, math enthusiasts! Ever stumbled upon an expression that looks like a jumbled mess of numbers and variables? Don't worry, we've all been there. Today, we're going to break down a seemingly complex expression and simplify it step by step. Our focus is on the expression (3a^2b7)(5a^3b8). This looks intimidating at first glance, but with a few basic rules of algebra, we can tame this beast. We'll walk you through the process, explaining each step clearly, so you can confidently tackle similar problems in the future. So, grab your pencils and notebooks, and let's dive into the world of exponents and coefficients!

Understanding the Basics

Before we jump into the solution, let's quickly recap some fundamental concepts. These are the building blocks that will help us understand how to simplify expressions like (3a^2b7)(5a^3b8). Remember, math is like a language; once you understand the grammar, you can speak it fluently!

• Coefficients: These are the numerical parts of a term (the numbers in front of the variables). In our expression, the coefficients are 3 and 5.
• Variables: These are the letters representing unknown values (like 'a' and 'b' in our case).
• Exponents: These are the small numbers written above and to the right of the variables. They indicate the power to which the variable is raised (e.g., in a^2, the exponent is 2, meaning 'a' is raised to the power of 2, or a * a).
• Terms: A term can be a single number, a single variable, or numbers and variables multiplied together. In the expression (3a^2b7)(5a^3b8), we have two terms within the parentheses.

The key rule we'll be using today is the product of powers rule. This rule states that when multiplying terms with the same base, you add their exponents. Mathematically, it looks like this: x^m * x^n = x^(m+n). This rule is the engine that will drive our simplification process, so make sure you have a good grasp of it. We'll see how this rule applies to our expression in the next section.

Breaking Down the Expression

Now, let's get our hands dirty and start simplifying (3a^2b7)(5a^3b8). The first thing we want to do is rearrange the terms to group like terms together. This makes the multiplication process much clearer and less prone to errors. Remember, the order in which we multiply numbers doesn't change the result (this is the commutative property of multiplication), so we can shuffle things around as needed. Our expression can be rewritten as:

(3 * 5) * (a^2 * a^3) * (b^7 * b^8)

See how we've grouped the coefficients together, the 'a' terms together, and the 'b' terms together? This is a crucial step in simplifying any algebraic expression. Now, we can deal with each group separately. Let's start with the coefficients. Multiplying 3 and 5 is straightforward: 3 * 5 = 15. So, we have simplified the coefficient part of our expression. Next, we move on to the variable terms. This is where the product of powers rule comes into play. For the 'a' terms, we have a^2 * a^3. According to the rule, we add the exponents: 2 + 3 = 5. So, a^2 * a^3 simplifies to a^5. Similarly, for the 'b' terms, we have b^7 * b^8. Adding the exponents, we get 7 + 8 = 15. Therefore, b^7 * b^8 simplifies to b^15. Now we have all the pieces of the puzzle. We've simplified the coefficients, the 'a' terms, and the 'b' terms. All that's left is to put them back together.

Putting It All Together

We've done the hard work of breaking down and simplifying each part of the expression (3a^2b7)(5a^3b8). Now comes the satisfying step of combining our results. We found that:

• 3 * 5 = 15
• a^2 * a^3 = a^5
• b^7 * b^8 = b^15

To get the final simplified expression, we simply multiply these results together. This gives us:

15 * a^5 * b^15

Which is usually written as:

15a^5b15

And there you have it! We've successfully simplified the original expression. Notice how much cleaner and easier to understand the simplified form is compared to the original. This is the power of algebraic simplification – it allows us to express complex relationships in a concise and manageable way. But our journey doesn't end here. Let's take a moment to reflect on the process we used and highlight the key takeaways. This will solidify your understanding and help you apply these techniques to other problems.

Key Takeaways and Practice

Let's recap the key steps we took to simplify the expression (3a^2b7)(5a^3b8). This will help you internalize the process and apply it to similar problems in the future. Remember, practice makes perfect, so don't be afraid to tackle more examples!

1. Rearrange the terms: Group the coefficients together and the variables with the same base together. This makes the multiplication process clearer.
2. Multiply the coefficients: This is usually a straightforward arithmetic operation.
3. Apply the product of powers rule: When multiplying variables with the same base, add their exponents (x^m * x^n = x^(m+n)).
4. Combine the results: Multiply the simplified coefficients and variable terms together to get the final simplified expression.

In our case, these steps led us to the solution 15a^5b15. Now, to solidify your understanding, try applying these steps to other similar expressions. For example, you could try simplifying (2x^3y2)(4x^2y5) or (7p^4q)(3p2q^3). The more you practice, the more comfortable you'll become with these concepts. Remember, simplifying expressions is a fundamental skill in algebra, and mastering it will open doors to more advanced topics. So, keep practicing, keep exploring, and keep enjoying the beauty of mathematics!

In conclusion, the simplified form of the expression (3a^2b7)(5a^3b8) is 15a^5b15. This involved multiplying the coefficients and applying the product of powers rule to the variables. For further learning on algebraic expressions and simplification, you can visit resources like Khan Academy's Algebra Section.