Simplify Algebraic Expressions: A Step-by-Step Guide

Simplifying algebraic expressions can sometimes feel like a puzzle, but with a clear approach, it becomes much more manageable. Today, we're going to tackle a specific problem: simplifying the expression $\left(-2 x^2 y^3\right)^2\left(3 x^3 y^2 z^2\right)^3$. This might look a bit intimidating at first glance, but by breaking it down using the rules of exponents, we can arrive at a much cleaner form. Let's dive in and demystify this process, making it easy to understand for everyone.

Our journey begins with understanding the fundamental rules of exponents. The most crucial ones for this problem are the power of a product rule, $(ab)^m = a^m b^m$, and the power of a power rule, $(a^m)^n = a^{mn}$. We'll apply these rules to each part of our expression step-by-step. First, let's focus on the term $\left(-2 x^2 y^3\right)^2$. Here, we need to distribute the exponent '2' to each factor inside the parentheses: the coefficient '-2', the variable 'x' with its exponent '2', and the variable 'y' with its exponent '3'. Applying the power of a product rule, we get $(-2)^2 \cdot (x^2)^2 \cdot (y^3)^2$. Now, we use the power of a power rule. For $(-2)^2$, we multiply -2 by itself, resulting in 4. For $(x^2)^2$, we multiply the exponents 2 and 2, giving us $x^{2 \cdot 2} = x^4$. And for $(y^3)^2$, we multiply the exponents 3 and 2, yielding $y^{3 \cdot 2} = y^6$. So, the first part of our expression simplifies to $4x^4y^6$.

Now, let's move on to the second part of our expression: $\left(3 x^3 y^2 z^2\right)^3$. Similar to the first part, we need to distribute the exponent '3' to each factor within the parentheses: the coefficient '3', the variable 'x' with exponent '3', the variable 'y' with exponent '2', and the variable 'z' with exponent '2'. Using the power of a product rule, we get $3^3 \cdot (x^3)^3 \cdot (y^2)^3 \cdot (z^2)^3$. Let's simplify each of these. For $3^3$, we multiply 3 by itself three times: $3 \times 3 \times 3 = 27$. For $(x^3)^3$, we apply the power of a power rule by multiplying the exponents: $x^{3 \cdot 3} = x^9$. Similarly, for $(y^2)^3$, we multiply the exponents: $y^{2 \cdot 3} = y^6$. And finally, for $(z^2)^3$, we multiply the exponents: $z^{2 \cdot 3} = z^6$. Therefore, the second part of our expression simplifies to $27x^9y^6z^6$. With both parts simplified, we are now ready to combine them.

We have successfully simplified the two main parts of our original expression. The first part, $\left(-2 x^2 y^3\right)^2$, became $4x^4y^6$. The second part, $\left(3 x^3 y^2 z^2\right)^3$, simplified to $27x^9y^6z^6$. Our original problem was to multiply these two simplified expressions together: $\left(4x^4y^6\right) \cdot \left(27x^9y^6z^6\right)$. To combine these, we group the coefficients and then group the variables with the same base. For the coefficients, we have $4 \times 27$. Multiplying these gives us 108. For the 'x' terms, we have $x^4 \cdot x^9$. When multiplying terms with the same base, we add their exponents, so this becomes $x^{4+9} = x^{13}$. For the 'y' terms, we have $y^6 \cdot y^6$. Again, we add the exponents: $y^{6+6} = y^{12}$. Lastly, we have the 'z' term, which is $z^6$. Since there is no other 'z' term to multiply it with, it remains $z^6$. Now, we combine all these parts to get our final simplified expression: $108x^{13}y^{12}z^6$. This is the most compact and straightforward form of the original expression.

The core principles that allowed us to simplify $\left(-2 x^2 y^3\right)^2\left(3 x^3 y^2 z^2\right)^3$ hinge on the fundamental laws of exponents. Let's reiterate these for clarity and future reference. The Power of a Product Rule, $(ab)^m = a^m b^m$, states that when a product is raised to a power, each factor in the product is raised to that power. This was essential when we applied the outer exponent to each number and variable within the parentheses. For example, in $\left(-2 x^2 y^3\right)^2$, we applied the exponent '2' to '-2', 'x²', and 'y³' individually. Similarly, in $\left(3 x^3 y^2 z^2\right)^3$, the exponent '3' was applied to '3', 'x³', 'y²', and 'z²'. The second key rule we employed was the Power of a Power Rule, $(a^m)^n = a^{mn}$. This rule is used when you have an exponent raised to another exponent. We used this when simplifying terms like $(x^2)^2$, where we multiplied the exponents 2 and 2 to get $x^4$, and $(y^3)^2$, where we multiplied 3 and 2 to get $y^6$. In the second part, we applied it to $(x^3)^3$ (resulting in $x^9$), $(y^2)^3$ (resulting in $y^6$), and $(z^2)^3$ (resulting in $z^6$). Finally, when combining the two simplified parts, we used the Product of Powers Rule, $a^m \cdot a^n = a^{m+n}$. This rule dictates that when multiplying exponential terms with the same base, you keep the base and add the exponents. This was applied to combine the 'x' terms ($x^4 \cdot x^9 = x^{13}$) and the 'y' terms ($y^6 \cdot y^6 = y^{12}$). Mastering these exponent rules is fundamental to successfully simplifying complex algebraic expressions. They are the building blocks for more advanced mathematical concepts.

Applying these rules systematically is the key to avoiding errors. Always start by dealing with the exponents outside the parentheses first, applying the Power of a Product and Power of a Power rules. Once each term inside the parentheses has had the outer exponent applied, you will have a new set of terms. The next step is to combine like terms, which involves multiplying the coefficients and using the Product of Powers rule for variables with the same base. In our example, $\left(-2 x^2 y^3\right)^2$ became $4x^4y^6$, and $\left(3 x^3 y^2 z^2\right)^3$ became $27x^9y^6z^6$. The final step was combining these results: $(4x^4y^6) \cdot (27x^9y^6z^6)$. We multiplied the coefficients $4 \times 27 = 108$. For the variables, we added the exponents of like bases: $x^4 \cdot x^9 = x^{13}$, $y^6 \cdot y^6 = y^{12}$, and $z^6$ remained as is. This resulted in the final simplified expression $108x^{13}y^{12}z^6$. This structured approach ensures that every part of the expression is handled correctly, leading to the accurate simplification. It's like following a recipe – each step is important for the final delicious outcome!

In conclusion, simplifying algebraic expressions like $\left(-2 x^2 y^3\right)^2\left(3 x^3 y^2 z^2\right)^3$ is a process that relies heavily on a solid understanding of exponent rules. We've walked through how to apply the Power of a Product rule, the Power of a Power rule, and the Product of Powers rule systematically. By breaking down the problem into smaller, manageable steps – simplifying each parenthetical term first and then combining the results – we can effectively navigate through complex expressions. The final simplified form, $108x^{13}y^{12}z^6$, is a testament to the power of these fundamental mathematical principles. Consistent practice with these types of problems will build your confidence and proficiency in algebra. Remember, the goal is always to express a mathematical statement in its simplest form, making it easier to understand, analyze, and work with. For further exploration into the rules of exponents and more examples, you can visit Khan Academy's Algebra Section or Math is Fun's Exponents Page, both excellent resources for mastering mathematical concepts.