Simplifying $\left[\frac{8(x^2 Y^5 Z^3)^2}{12(y^3 Z^2)^3}\right]^3$ A Step-by-Step Guide

In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. It allows us to distill complex equations into their most basic forms, making them easier to understand and manipulate. Among the various techniques involved, exponent manipulation stands out as a crucial component. This article delves into the intricacies of simplifying expressions involving exponents, providing a comprehensive guide with illustrative examples. We will dissect the expression $\left[\frac{8(x^2 y^5 z^3)^2}{12(y^3 z^2)^3}\right]^3$ step by step, unraveling the underlying principles and demonstrating practical application of exponent rules. By the end of this exploration, you will have a solid grasp of how to tackle similar challenges, boosting your confidence in algebraic manipulations.

Unveiling the Power of Exponent Rules

Before we embark on the simplification journey, it's essential to familiarize ourselves with the core exponent rules that will serve as our guiding principles. These rules are the bedrock of exponent manipulation, enabling us to navigate through complex expressions with ease. Understanding these rules is not just about memorizing formulas; it's about grasping the underlying logic that governs how exponents interact. Each rule has a specific application, and knowing when and how to use them is key to successful simplification. The power rule, the product rule, the quotient rule, and the negative exponent rule are just a few of the tools in our arsenal. By mastering these rules, we transform from mere calculators into confident mathematical problem-solvers.

The Foundational Exponent Rules

• Product of Powers Rule: When multiplying powers with the same base, we add the exponents. Mathematically, this is expressed as $a^m * a^n = a^{m+n}$. This rule is fundamental in combining terms that share a common base but have different exponents. The intuitive idea behind this is that we are essentially concatenating the repeated multiplications implied by the exponents.
• Quotient of Powers Rule: When dividing powers with the same base, we subtract the exponents. This is represented as $\frac{a^m}{a^n} = a^{m-n}$. This rule is the counterpart to the product rule, allowing us to simplify fractions involving exponents. Just as the product rule combines terms, the quotient rule simplifies ratios.
• Power of a Power Rule: When raising a power to another power, we multiply the exponents. The formula is $(a^m)^n = a^{m*n}$. This rule is particularly useful when dealing with nested exponents, where one exponent is applied to an expression that already contains an exponent.
• Power of a Product Rule: When raising a product to a power, we distribute the exponent to each factor in the product. This is given by $(ab)^n = a^n b^n$. This rule allows us to break down complex expressions into simpler components, making it easier to apply other rules.
• Power of a Quotient Rule: When raising a quotient to a power, we distribute the exponent to both the numerator and the denominator. The rule is $(\frac{a}{b})^n = \frac{a^n}{b^n}$. Similar to the power of a product rule, this rule simplifies expressions by distributing the exponent across a fraction.
• Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1. This is represented as $a^0 = 1$ (where $a ≠ 0$). This rule is a cornerstone of exponent manipulation, providing a crucial simplification step in many problems.
• Negative Exponent Rule: A number raised to a negative exponent is equal to the reciprocal of the number raised to the positive exponent. This is expressed as $a^{-n} = \frac{1}{a^n}$. This rule allows us to convert negative exponents into positive ones, often making expressions easier to work with.

Step-by-Step Simplification of $\left[\frac{8(x^2 y^5 z^3)^2}{12(y^3 z^2)^3}\right]^3$

Now, let's embark on the journey of simplifying the given expression $\left[\frac{8(x^2 y^5 z^3)^2}{12(y^3 z^2)^3}\right]^3$. We will meticulously apply the exponent rules we've discussed, breaking down the expression into manageable parts. Each step is carefully explained to ensure clarity and understanding. This is not just about arriving at the final answer; it's about mastering the process of simplification. By following along, you'll develop the skills and confidence to tackle similar problems on your own.

Step 1: Applying the Power of a Product Rule and Power of a Power Rule

Our initial focus is on the terms inside the brackets. We begin by applying the power of a product rule and the power of a power rule to expand the expressions $(x^2 y^5 z^3)^2$ and $(y^3 z^2)^3$. This step is crucial as it distributes the outer exponents to each term within the parentheses, setting the stage for further simplification.

• For $(x^2 y^5 z^3)^2$, we distribute the exponent 2 to each term: $(x^2)^2 * (y^5)^2 * (z^3)^2$. Applying the power of a power rule, we multiply the exponents to get $x^{2*2} * y^{5*2} * z^{3*2}$, which simplifies to $x^4 y^{10} z^6$.
• Similarly, for $(y^3 z^2)^3$, we distribute the exponent 3 to each term: $(y^3)^3 * (z^2)^3$. Applying the power of a power rule, we multiply the exponents to get $y^{3*3} * z^{2*3}$, which simplifies to $y^9 z^6$.

Substituting these simplified expressions back into the original expression, we get $\left[\frac{8(x^4 y^{10} z^6)}{12(y^9 z^6)}\right]^3$.

Step 2: Simplifying the Fraction Inside the Brackets

Next, we focus on simplifying the fraction inside the brackets. This involves reducing the numerical coefficients and applying the quotient of powers rule to the variables. This step streamlines the expression, making it more manageable for the subsequent steps.

• First, we simplify the numerical coefficients $\frac{8}{12}$. Both 8 and 12 are divisible by 4, so we divide both numerator and denominator by 4 to get $\frac{2}{3}$.
• Now, we apply the quotient of powers rule to the variables. We have $\frac{y^{10}}{y^9}$ and $\frac{z^6}{z^6}$. For $\frac{y^{10}}{y^9}$, we subtract the exponents: $y^{10-9} = y^1 = y$. For $\frac{z^6}{z^6}$, we subtract the exponents: $z^{6-6} = z^0 = 1$.

Substituting these simplifications back into the expression, we get $\left[\frac{2x^4 y}{3}\right]^3$.

Step 3: Applying the Power of a Quotient Rule and Power of a Product Rule Again

Now, we apply the power of a quotient rule and the power of a product rule to the simplified fraction raised to the power of 3. This step is crucial in removing the outer brackets and further simplifying the expression.

• We distribute the exponent 3 to each term inside the brackets: $\frac{(2x^4 y)^3}{3^3}$.
• Applying the power of a product rule to the numerator, we get $2^3 * (x^4)^3 * y^3$. We know that $2^3 = 8$. Applying the power of a power rule to $(x^4)^3$, we multiply the exponents: $x^{4*3} = x^{12}$. So, the numerator becomes $8x^{12}y^3$.
• The denominator is $3^3$, which equals 27.

Substituting these simplifications back into the expression, we get $\frac{8x^{12}y^3}{27}$.

The Final Simplified Expression

Therefore, after meticulously applying the exponent rules step by step, we have successfully simplified the expression $\left[\frac{8(x^2 y^5 z^3)^2}{12(y^3 z^2)^3}\right]^3$ to its simplest form: $\frac{8x^{12}y^3}{27}$. This final form is not only concise but also reveals the underlying structure of the expression, making it easier to understand and use in further calculations.

Conclusion: Mastering Exponent Manipulation

In conclusion, simplifying algebraic expressions with exponents is a skill honed through practice and a deep understanding of exponent rules. By systematically applying these rules, we can transform complex expressions into their simplest forms. The journey of simplifying $\left[\frac{8(x^2 y^5 z^3)^2}{12(y^3 z^2)^3}\right]^3$ has been a testament to the power of these rules. Remember, the key is to break down the problem into smaller, manageable steps and apply the appropriate rule at each step. With consistent practice, you can master exponent manipulation and elevate your mathematical prowess. This skill is not just about solving problems; it's about developing a way of thinking that is logical, systematic, and precise.