Evaluating (a^2 B)^-2(a C^2)^2 ⋅ (b^2 C)^3 Given A=-1, B=1, And C=2 A Step-by-Step Guide

Jun 16, 2025 by ADMIN 89 views

$Evaluating the Expression (a^2 b)^{-2}(a c^2)^2 ⋅ (b^2 c)^3 Given a=-1, b=1, and c=2$

This article provides a step-by-step guide to evaluating the algebraic expression $\left(a^2 b\right)^{-2}\left(a c$ when $a=-1$ , $b=1$ , and $c=2$ . We will walk through the process of substituting the given values, simplifying the expression using exponent rules, and arriving at the final numerical answer. This problem combines the concepts of algebraic expressions, exponent rules, and substitution, making it an excellent exercise for those looking to strengthen their foundational mathematics skills.

1. Understanding the Problem

Before diving into the solution, let's clearly define the problem. We are given an algebraic expression involving variables a, b, and c, and specific numerical values for each variable. Our goal is to find the value of the entire expression when we replace the variables with their corresponding values. This requires a careful application of the order of operations and the rules of exponents.

Specifically, we are tasked with finding the value of $\left(a^2 b\right)^{-2}\left(a c^2\right)^2 \cdot\left(b^2 c\right)^3$ when $a=-1$ , $b=1$ , and $c=2$ . To solve this, we will first substitute the values of $a$ , $b$ , and $c$ into the expression. Then, we will simplify the expression using the rules of exponents, such as the power of a product rule, the power of a power rule, and the negative exponent rule. Finally, we will perform the arithmetic operations to obtain the final numerical value.

This problem is a great example of how algebra and arithmetic work together. We use algebraic principles to manipulate the expression and then use arithmetic to calculate the final result. By understanding the underlying concepts and following a systematic approach, we can confidently solve this type of problem.

2. Step-by-Step Solution

Now, let's embark on the step-by-step solution to the problem. This section will meticulously detail each step, ensuring clarity and understanding. We'll begin by substituting the values of a, b, and c into the expression, and then progressively simplify the expression using exponent rules.

2.1 Substitution

The first step is to substitute the given values of $a = -1$ , $b = 1$ , and $c = 2$ into the expression $\left(a^2 b\right)^{-2}\left(a c^2\right)^2 \cdot\left(b^2 c\right)^3$ . This gives us:

$\left((-1)^2 (1)\right)^{-2}\left((-1) (2)^2\right)^2 \cdot\left((1)^2 (2)\right)^3$

This substitution replaces each variable with its numerical value, setting the stage for simplification. It's crucial to perform this step carefully, ensuring that each variable is replaced correctly. This is the foundation upon which the rest of the solution is built.

2.2 Simplifying within Parentheses

Next, we simplify the expressions inside the parentheses, following the order of operations (PEMDAS/BODMAS). We start by evaluating the exponents within the parentheses:

$\left((1)(1)\right)^{-2}\left((-1)(4)\right)^2 \cdot\left((1)(2)\right)^3$

Here, we've calculated $(-1)^2 = 1$ , $(2)^2 = 4$ , and $(1)^2 = 1$ . Now, we perform the multiplications within the parentheses:

$\left(1\right)^{-2}\left(-4\right)^2 \cdot\left(2\right)^3$

This step simplifies the expression within the parentheses, making it easier to apply the exponent rules in the next step. The focus is on reducing the complexity of the terms inside the parentheses before moving on to the outer exponents.

2.3 Applying Exponent Rules

Now, we apply the exponent rules to simplify the expression further. We have three terms, each raised to a power. Let's evaluate each term individually:

$\left(1\right)^{-2} = 1$ (Any power of 1 is 1)
$\left(-4\right)^2 = 16$ (Squaring -4 gives 16)
$\left(2\right)^3 = 8$ (2 cubed is 8)

Substituting these values back into the expression, we get:

$(1)(16)(8)$

This step is crucial as it utilizes the fundamental rules of exponents to simplify the terms. Understanding and applying these rules correctly is essential for solving this type of problem.

2.4 Final Calculation

The final step is to perform the multiplication:

$(1)(16)(8) = 128$

Therefore, the value of the expression $\left(a^2 b\right)^{-2}\left(a c^2\right)^2 \cdot\left(b^2 c\right)^3$ when $a=-1$ , $b=1$ , and $c=2$ is 128.

This final calculation brings us to the solution. By systematically working through each step, from substitution to simplification, we have successfully evaluated the given expression.

3. Key Concepts and Rules Used

Throughout the solution, we utilized several key mathematical concepts and rules. Understanding these principles is crucial for solving similar problems. Let's recap the main concepts applied:

Substitution: Replacing variables with their given numerical values is a fundamental technique in algebra. It allows us to evaluate expressions and solve equations for specific scenarios.
Order of Operations (PEMDAS/BODMAS): The order of operations dictates the sequence in which operations are performed in a mathematical expression. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This ensures consistent and accurate calculations.
Exponent Rules: These rules govern how exponents interact with different operations. The key rules used in this problem include:
- Power of a Product: $(ab)^n = a^n b^n$ (The power of a product is the product of the powers)
- Power of a Power: $(a^m)^n = a^{mn}$ (A power raised to another power is the base raised to the product of the powers)
- Negative Exponent: $a^{-n} = \frac{1}{a^n}$ (A negative exponent indicates the reciprocal of the base raised to the positive exponent)
- Any power of 1: $1^n = 1$ (Any power of 1 is 1)

Understanding these concepts and rules is essential for mastering algebraic manipulations and problem-solving.

4. Common Mistakes to Avoid

When solving problems like this, it's easy to make small errors that can lead to an incorrect answer. Being aware of common pitfalls can help you avoid them. Here are some common mistakes to watch out for:

Incorrect Substitution: A common error is substituting the values of the variables incorrectly. Double-check that you've replaced each variable with its correct value.
Order of Operations Errors: Failing to follow the order of operations (PEMDAS/BODMAS) can lead to incorrect results. Make sure you perform operations in the correct sequence.
Exponent Rule Mistakes: Incorrectly applying exponent rules is a frequent source of errors. Review the exponent rules carefully and make sure you understand how they work.
Sign Errors: Dealing with negative numbers can be tricky. Pay close attention to signs, especially when squaring negative numbers or applying the negative exponent rule.
Arithmetic Errors: Simple arithmetic mistakes can happen, especially when dealing with multiple operations. Double-check your calculations to minimize errors.

By being mindful of these common mistakes and taking the time to check your work, you can significantly reduce the chances of making errors.

5. Practice Problems

To solidify your understanding of evaluating algebraic expressions, here are a few practice problems:

Evaluate $\left(x^3 y\right)^{-1} \left(x y^2\right)^2$ when $x=2$ and $y=-1$ .
Find the value of $\frac{\left(p^2 q^{-1}\right)^3}{\left(p^{-1} q^2\right)^2}$ if $p=-2$ and $q=3$ .
Simplify and evaluate $\left(m^{-2} n^3\right)^2 \cdot \left(m^4 n^{-1}\right)$ when $m=1$ and $n=-2$ .

Working through these problems will help you gain confidence in your ability to apply the concepts and techniques discussed in this article. Remember to follow the step-by-step approach, pay attention to the order of operations, and be mindful of potential errors.

6. Conclusion

In conclusion, evaluating the expression $\left(a^2 b\right)^{-2}\left(a c^2\right)^2 \cdot\left(b^2 c\right)^3$ when $a=-1$ , $b=1$ , and $c=2$ demonstrates the importance of understanding and applying algebraic principles, exponent rules, and the order of operations. By meticulously following a step-by-step approach, we successfully simplified the expression and arrived at the solution, 128.

This type of problem highlights the interconnectedness of different mathematical concepts. Substitution, exponent rules, and arithmetic operations all play a crucial role in arriving at the final answer. Mastering these concepts is essential for success in algebra and beyond.

Remember, practice is key to improving your mathematical skills. By working through similar problems and being mindful of common mistakes, you can build your confidence and proficiency in evaluating algebraic expressions. The practice problems provided offer an excellent opportunity to further refine your understanding and skills in this area.