Find The Value Of (ab^2)^{-2}(a^2 C)^2 (bc^2)^2

In this article, we will delve into the realm of algebraic expressions and explore how to simplify and evaluate them. Our focus will be on a specific expression: $(ab2){-2}(a^2 c)^2 (bc2)2$, where we are given the values of the variables $a$, $b$, and $c$ as $1$, $2$, and $1$ respectively. We will systematically break down the expression, applying the rules of exponents and algebraic manipulation to arrive at the final numerical value. This exercise not only reinforces our understanding of algebraic principles but also demonstrates the power of these principles in solving mathematical problems.

Understanding the Fundamentals of Exponents

Before we embark on the journey of simplifying the given expression, it's crucial to recapitulate the fundamental laws of exponents. These laws serve as the bedrock of our algebraic manipulations and will guide us through the simplification process. Let's delve into some key exponent rules that we will be employing:

1. Product of Powers Rule: This rule states that when multiplying exponents with the same base, we add the powers. Mathematically, it's represented as $x^m * x^n = x^{m+n}$. For instance, if we have $2^3 * 2^2$, we can simplify it as $2^{3+2} = 2^5 = 32$.

2. Quotient of Powers Rule: Conversely, when dividing exponents with the same base, we subtract the powers. This can be expressed as $x^m / x^n = x^{m-n}$. For example, $3^5 / 3^2$ simplifies to $3^{5-2} = 3^3 = 27$.

3. Power of a Power Rule: When raising a power to another power, we multiply the exponents. The rule is represented as $(x^m)^n = x^{m*n}$. Consider $(4^2)^3$, which simplifies to $4^{2*3} = 4^6 = 4096$.

4. Power of a Product Rule: This rule states that when raising a product to a power, we raise each factor in the product to that power. The rule is $(xy)^n = x^n y^n$. An example is $(2 * 3)^2$, which can be simplified as $2^2 * 3^2 = 4 * 9 = 36$.

5. Power of a Quotient Rule: Similarly, when raising a quotient to a power, we raise both the numerator and the denominator to that power. The rule is $(x/y)^n = x^n / y^n$. For instance, $(5/2)^3$ simplifies to $5^3 / 2^3 = 125 / 8$.

6. Negative Exponent Rule: A negative exponent indicates the reciprocal of the base raised to the positive exponent. The rule is $x^{-n} = 1 / x^n$. For example, $2^{-3}$ is equivalent to $1 / 2^3 = 1 / 8$.

7. Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1. The rule is $x^0 = 1$ (where $x$ is not equal to 0). For example, $5^0 = 1$.

These exponent rules are the building blocks of simplifying algebraic expressions. Mastering these rules is essential for tackling more complex problems in algebra and beyond. As we proceed with the simplification of the given expression, we will be invoking these rules to break down the expression into manageable components and ultimately arrive at the final answer.

Breaking Down the Expression: A Step-by-Step Approach

Now that we have a firm grasp of the exponent rules, let's embark on the task of simplifying the expression $(ab^2)^{-2}(a^2 c)^2 (bc^2)^2$. We will adopt a systematic, step-by-step approach to ensure clarity and accuracy. This will involve applying the exponent rules judiciously and simplifying the expression iteratively until we reach its most concise form.

Step 1: Distribute the Outer Exponents

Our initial focus will be on dealing with the exponents outside the parentheses. We will employ the Power of a Product Rule and the Power of a Power Rule to distribute these exponents to each factor within the parentheses. This will help us to break down the expression into smaller, more manageable components.

• For the term $(ab^2)^{-2}$, we distribute the exponent -2 to both $a$ and $b^2$. This gives us $a^{-2} (b^2)^{-2}$, which further simplifies to $a^{-2} b^{-4}$.
• Next, we consider the term $(a^2 c)^2$. Distributing the exponent 2, we get $(a^2)^2 c^2$, which simplifies to $a^4 c^2$.
• Finally, we address the term $(bc^2)^2$. Distributing the exponent 2, we obtain $b^2 (c^2)^2$, which simplifies to $b^2 c^4$.

Now, our expression looks like this: $a^{-2} b^{-4} * a^4 c^2 * b^2 c^4$.

Step 2: Combine Like Terms

Having distributed the exponents, our next objective is to consolidate the terms with the same base. This involves using the Product of Powers Rule, which states that when multiplying exponents with the same base, we add the powers. We will group together the terms with the same base and then apply this rule to simplify them.

• Let's start with the terms involving $a$. We have $a^{-2}$ and $a^4$. Applying the Product of Powers Rule, we get $a^{-2} * a^4 = a^{-2+4} = a^2$.
• Next, we consider the terms involving $b$. We have $b^{-4}$ and $b^2$. Applying the Product of Powers Rule, we get $b^{-4} * b^2 = b^{-4+2} = b^{-2}$.
• Finally, we address the terms involving $c$. We have $c^2$ and $c^4$. Applying the Product of Powers Rule, we get $c^2 * c^4 = c^{2+4} = c^6$.

After combining like terms, our expression simplifies to $a^2 b^{-2} c^6$.

Step 3: Eliminate Negative Exponents

In general, it's considered good practice to express our final answer without any negative exponents. To achieve this, we will employ the Negative Exponent Rule, which states that $x^{-n} = 1 / x^n$. We will identify any terms with negative exponents and rewrite them using this rule.

In our expression, we have the term $b^{-2}$, which has a negative exponent. Applying the Negative Exponent Rule, we can rewrite this as $1 / b^2$.

Therefore, our expression now becomes $a^2 * (1 / b^2) * c^6$, which can be written as $(a^2 c^6) / b^2$.

We have now successfully simplified the expression $(ab^2)^{-2}(a^2 c)^2 (bc^2)^2$ to its simplest form: $(a^2 c^6) / b^2$. This step-by-step approach, employing the exponent rules, has allowed us to transform the complex initial expression into a much more manageable form.

Substituting the Values and Calculating the Result

Having simplified the expression to $(a^2 c^6) / b^2$, the next step is to substitute the given values of $a$, $b$, and $c$ into this simplified expression. This will allow us to evaluate the expression and obtain a numerical result. We are given that $a = 1$, $b = 2$, and $c = 1$. Let's substitute these values into our simplified expression:

$(a^2 c^6) / b^2 = (1^2 * 1^6) / 2^2$

Now, we need to evaluate the powers:

• $1^2 = 1 * 1 = 1$
• $1^6 = 1 * 1 * 1 * 1 * 1 * 1 = 1$
• $2^2 = 2 * 2 = 4$

Substituting these values back into the expression, we get:

$(1 * 1) / 4 = 1 / 4$

Therefore, the final value of the expression $(ab^2)^{-2}(a^2 c)^2 (bc^2)^2$ when $a = 1$, $b = 2$, and $c = 1$ is $1/4$ or $0.25$. This completes our evaluation of the algebraic expression, demonstrating the power of simplification and substitution in solving mathematical problems.

Conclusion: The Power of Algebraic Simplification

In conclusion, we have successfully navigated the process of simplifying and evaluating the algebraic expression $(ab^2)^{-2}(a^2 c)^2 (bc^2)^2$. By systematically applying the rules of exponents and employing a step-by-step approach, we transformed the complex initial expression into a much simpler form: $(a^2 c^6) / b^2$. Subsequently, by substituting the given values of $a = 1$, $b = 2$, and $c = 1$, we were able to calculate the numerical value of the expression, which turned out to be $1/4$ or $0.25$.

This exercise underscores the importance of algebraic simplification in mathematics. Simplification not only makes expressions more manageable but also reveals the underlying structure and relationships within them. Moreover, it facilitates the process of evaluation, allowing us to arrive at concrete numerical results.

The exponent rules, which served as the foundation of our simplification process, are fundamental tools in algebra. Mastering these rules is essential for tackling a wide range of mathematical problems, from basic algebraic manipulations to more advanced concepts in calculus and beyond. The ability to confidently apply these rules empowers us to solve complex problems with ease and efficiency.

Furthermore, the step-by-step approach we adopted highlights the value of methodical problem-solving. By breaking down the problem into smaller, more manageable steps, we were able to avoid errors and maintain clarity throughout the process. This approach is applicable not only in mathematics but also in various other domains, emphasizing the importance of structured thinking in problem-solving.

In essence, this exercise demonstrates the power of algebraic manipulation in conjunction with the fundamental rules of exponents. By mastering these techniques and adopting a systematic approach, we can confidently tackle a wide array of mathematical challenges and gain a deeper appreciation for the elegance and power of mathematics.