Simplifying Algebraic Expressions: A Step-by-Step Guide

Hey guys! Let's dive into the world of algebra and learn how to simplify the expression $\left(-3 y^2 z^3\right)^3\left(x^2 y^4 z\right)$. It might look a bit intimidating at first, but trust me, it's just a matter of applying a few simple rules. This detailed guide will break down the process step-by-step, making it super easy to understand and master. We'll cover everything from the basics of exponents to combining like terms, ensuring you have a solid grasp of simplifying algebraic expressions. Ready to get started? Let's go!

Understanding the Basics: Exponents and Their Rules

Before we jump into the simplification, let's brush up on the fundamental concepts. At the heart of this problem are exponents, those little numbers that sit above and to the right of a variable or number. They tell us how many times to multiply the base by itself. For example, $x^2$ means $x$ multiplied by itself twice ($x * x$), and $y^3$ means $y$ multiplied by itself three times ($y * y * y$).

The most important rule to remember here is the power of a product rule: $(ab)^n = a^n * b^n$. This rule says that when you have a product raised to a power, you can distribute the exponent to each factor within the parentheses. For instance, $(2x)^3 = 2^3 * x^3 = 8x^3$. Another critical rule is the power of a power rule: $(a^m)^n = a^{m*n}$. This means that when you have a power raised to another power, you multiply the exponents. For example, $(x^2)^3 = x^(2*3) = x^6$. Understanding these rules is crucial because we'll be using them extensively in our simplification process.

Now, let's talk about negative numbers and exponents. When you have a negative number raised to an odd power, the result is negative. For example, $(-2)^3 = -8$. But, when you have a negative number raised to an even power, the result is positive. For example, $(-2)^2 = 4$. This is important to remember when dealing with the coefficient $-3$ in our original expression. Also, when multiplying terms with the same base, you add the exponents. For instance, $x^2 * x^3 = x^(2+3) = x^5$. Keep these rules in mind, and you'll be well-equipped to tackle our expression.

Breaking Down the Expression: The First Steps

Alright, let's get our hands dirty with the expression $\left(-3 y^2 z^3\right)^3\left(x^2 y^4 z\right)$. The first thing we need to do is address the exponent outside the first set of parentheses. We're dealing with $\left(-3 y^2 z^3\right)^3$. Using the power of a product rule, we need to distribute the exponent $3$ to each term inside the parentheses. This means cubing $-3$, cubing $y^2$, and cubing $z^3$. Let's break it down:

• $(-3)^3 = -27$ (remember, a negative number raised to an odd power is negative)
• $(y^2)^3 = y^(2*3) = y^6$ (using the power of a power rule)
• $(z^3)^3 = z^(3*3) = z^9$ (again, using the power of a power rule)

So, after simplifying the first part, $\left(-3 y^2 z^3\right)^3$ becomes $-27y^6z^9$. Now, our expression looks like this: $-27y^6z^9\left(x^2 y^4 z\right)$. See? We're already making progress! The key is to take it step by step and apply the rules meticulously. Don't rush; take your time, and you'll get it.

Combining Like Terms: The Final Push

Now that we've simplified the first part of the expression, it's time to combine like terms. Remember, like terms are terms that have the same variables raised to the same powers. In our simplified expression, $-27y^6z^9\left(x^2 y^4 z\right)$, we can multiply $-27y^6z^9$ by $x^2 y^4 z$. Let's go through it carefully.

First, multiply the coefficients: $-27 * 1 = -27$. Then, let's handle the variables one by one:

• $x^2$ is the only $x$ term, so it remains as $x^2$.
• For the $y$ terms, we have $y^6$ and $y^4$. When multiplying like terms, we add the exponents: $y^6 * y^4 = y^(6+4) = y^{10}$.
• For the $z$ terms, we have $z^9$ and $z^1$ (remember, $z$ is the same as $z^1$). Again, add the exponents: $z^9 * z^1 = z^(9+1) = z^{10}$.

Putting it all together, we get $-27x^2y^{10}z^{10}$. And that's it, guys! We've successfully simplified the expression. The final answer is $-27x^2y^{10}z^{10}$.

Final Answer and Key Takeaways

So, to recap, the simplified form of $\left(-3 y^2 z^3\right)^3\left(x^2 y^4 z\right)$ is $\bf{-27x^2y^{10}z^{10}}$. We achieved this by following a few simple steps:

1. Applying the power of a product rule: We distributed the exponent to each term inside the parentheses. We've used the power of a product rule: $(ab)^n = a^n * b^n$.
2. Using the power of a power rule: We simplified the exponents by multiplying them: $(a^m)^n = a^{m*n}$.
3. Multiplying the coefficients: We multiplied the numerical factors.
4. Combining like terms: We multiplied the variables by adding their exponents when the bases were the same. The product rule for exponents. $x^m * x^n = x^(m+n)$.

The key takeaways here are understanding and applying the rules of exponents. Make sure you're comfortable with the power of a product, power of a power, and how to handle negative exponents. Also, practice combining like terms. With consistent practice, these concepts will become second nature, and you'll be simplifying algebraic expressions like a pro! Keep practicing, and don't be afraid to ask for help if you get stuck. You've got this!

This guide provided a detailed and easy-to-follow explanation of how to simplify the expression. With the basic rules of exponents and a step-by-step approach, simplifying algebraic expressions becomes a straightforward process. Keep practicing, and you'll gain confidence and proficiency in no time. Good job! Keep it up!