Simplify (x^9 Y)(x^{10} Y^7) With The Product Rule

Hey guys! Today, we're diving into the fascinating world of exponents and how to simplify expressions using the product rule. This is a fundamental concept in algebra, and mastering it will make your mathematical journey a whole lot smoother. We'll break down the process step-by-step, making sure you grasp every detail. So, let's jump right in and tackle the expression: $\left(x^9 y\right)\left(x^{10} y^7\right)$.

Understanding the Product Rule of Exponents

Before we dive into the specific problem, let's quickly recap the product rule of exponents. In essence, the product rule states that when you're multiplying terms with the same base, you add their exponents. Mathematically, this is expressed as: $a^m * a^n = a^{m+n}$. This rule is the cornerstone of simplifying expressions like the one we have. Remember, the product rule only applies when the bases are the same. For instance, you can apply the product rule to $x^9$ and $x^{10}$ because both terms have the base 'x'. Similarly, you can apply it to 'y' and $y^7$ since they share the base 'y'. However, you cannot directly apply the product rule to terms with different bases, like 'x' and 'y'. Think of it as combining like terms, but in the world of exponents! The product rule is not just a mathematical trick; it's a reflection of how exponents work. An exponent tells you how many times to multiply the base by itself. So, $x^9$ means x multiplied by itself nine times, and $x^{10}$ means x multiplied by itself ten times. When you multiply $x^9$ and $x^{10}$, you're essentially multiplying x by itself a total of 19 times (9 + 10), which is why the product rule says we add the exponents. Grasping this fundamental idea will make the product rule feel intuitive rather than just a formula to memorize. It's all about understanding the underlying concept of repeated multiplication that exponents represent.

Applying the Product Rule to the Given Expression

Now, let's apply the product rule to our expression: $\left(x^9 y\right)\left(x^{10} y^7\right)$. The first step is to identify the terms with the same base. We have $x^9$ and $x^{10}$ both with the base 'x', and 'y' (which can be thought of as $y^1$) and $y^7$ both with the base 'y'. Next, we apply the product rule to each pair of terms with the same base. For the 'x' terms, we have $x^9 * x^{10}$. According to the product rule, we add the exponents: 9 + 10 = 19. So, $x^9 * x^{10}$ simplifies to $x^{19}$. For the 'y' terms, we have $y * y^7$, which is the same as $y^1 * y^7$. Again, we add the exponents: 1 + 7 = 8. So, $y^1 * y^7$ simplifies to $y^8$. Now, we combine the simplified 'x' term and the simplified 'y' term. We have $x^{19}$ and $y^8$. Multiplying these together, we get our final simplified expression: $x^{19}y^8$. See how the product rule neatly combines the exponents of like bases? This is the power and elegance of this rule. By breaking down the expression into its components and applying the product rule to each pair of terms with the same base, we've successfully simplified the expression. Remember, it's all about adding the exponents when multiplying terms with the same base. Practice this a few times, and it'll become second nature!

Step-by-Step Solution

Let's break down the solution step-by-step to make it super clear:

1. Identify terms with the same base: In the expression $\left(x^9 y\right)\left(x^{10} y^7\right)$, we see $x^9$ and $x^{10}$ share the base 'x', and 'y' and $y^7$ share the base 'y'.
2. Apply the product rule to the 'x' terms: We have $x^9 * x^{10}$. Adding the exponents, 9 + 10 = 19. So, this simplifies to $x^{19}$.
3. Apply the product rule to the 'y' terms: We have $y * y^7$, which is the same as $y^1 * y^7$. Adding the exponents, 1 + 7 = 8. So, this simplifies to $y^8$.
4. Combine the simplified terms: We now have $x^{19}$ and $y^8$. Multiplying these together gives us the final simplified expression.

Therefore, $\left(x^9 y\right)\left(x^{10} y^7\right) = x^{19}y^8$.

This step-by-step approach makes it easy to follow the logic and ensures that you understand each step involved in simplifying the expression using the product rule. Remember, practice makes perfect! The more you work through problems like this, the more confident you'll become in applying the product rule and other exponent rules.

Common Mistakes to Avoid

When working with the product rule, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them. One frequent error is trying to apply the product rule to terms with different bases. Remember, the product rule only works when the bases are the same. For example, you can't directly simplify $x^9 * y^7$ using the product rule because 'x' and 'y' are different bases. Another common mistake is forgetting to add the exponents. The product rule specifically states that you add the exponents when multiplying terms with the same base. So, $x^9 * x^{10}$ becomes $x^{19}$ (9 + 10), not $x^{90}$ (9 * 10). Pay close attention to the operation! A third mistake is overlooking implied exponents. For instance, in the expression $\left(x^9 y\right)\left(x^{10} y^7\right)$, the 'y' term is implicitly $y^1$. Forgetting this '1' can lead to errors when adding the exponents. Remember to treat 'y' as $y^1$ when applying the product rule. Finally, some students might get confused about when to apply the product rule versus other exponent rules, like the power rule (which deals with raising a power to another power). The key is to recognize that the product rule applies when you are multiplying terms with the same base. Keeping these common mistakes in mind and practicing applying the product rule correctly will significantly improve your accuracy and understanding.

Practice Problems

To really solidify your understanding of the product rule, let's try a few practice problems. Working through these will help you gain confidence and identify any areas where you might need further review.

1. Simplify: $(a^5 b^2)(a^3 b^4)$
2. Simplify: $(2x^2 y^3)(3x^4 y)$
3. Simplify: $(c^7 d)(c^2 d^5)$

For the first problem, $(a^5 b^2)(a^3 b^4)$, identify the terms with the same base: $a^5$ and $a^3$, and $b^2$ and $b^4$. Apply the product rule to the 'a' terms: $a^5 * a^3 = a^{5+3} = a^8$. Apply the product rule to the 'b' terms: $b^2 * b^4 = b^{2+4} = b^6$. Combine the simplified terms: $a^8b^6$. So, the simplified expression is $a^8b^6$.

For the second problem, $(2x^2 y^3)(3x^4 y)$, we have numerical coefficients as well. Multiply the coefficients: 2 * 3 = 6. Apply the product rule to the 'x' terms: $x^2 * x^4 = x^{2+4} = x^6$. Apply the product rule to the 'y' terms: $y^3 * y = y^{3+1} = y^4$. Combine all the parts: 6 * $x^6$ * $y^4$. So, the simplified expression is $6x^6y^4$.

For the third problem, $(c^7 d)(c^2 d^5)$, apply the product rule to the 'c' terms: $c^7 * c^2 = c^{7+2} = c^9$. Apply the product rule to the 'd' terms: $d * d^5 = d^{1+5} = d^6$. Combine the simplified terms: $c^9d^6$. So, the simplified expression is $c^9d^6$.

Working through these examples helps illustrate how the product rule is applied in different contexts. Remember to always identify terms with the same base and then add their exponents. And don't forget to account for coefficients when they are present! Keep practicing, and you'll become a pro at simplifying expressions with the product rule.

Conclusion

So there you have it, guys! We've successfully simplified the expression $\left(x^9 y\right)\left(x^{10} y^7\right)$ using the product rule of exponents. Remember, the key is to identify terms with the same base and then add their exponents. We arrived at the simplified expression: $x^{19}y^8$. Mastering the product rule is a crucial step in your algebraic journey. It's a fundamental concept that pops up in many areas of mathematics, so understanding it well will pay dividends down the road. Keep practicing, and you'll be simplifying expressions like a champ in no time! And remember, math can be fun, especially when you break it down step-by-step and understand the underlying concepts. Keep exploring, keep learning, and most importantly, keep practicing!