Evaluating Algebraic Expressions Solving A³ - B² When A = -7 And B = -3

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Hey guys! Let's dive into the world of algebraic expressions and learn how to evaluate them with confidence. In this article, we'll break down a specific problem step by step, ensuring you grasp the fundamental concepts along the way. Whether you're a student tackling algebra for the first time or just looking to brush up your skills, this guide is for you. We'll take the expression a³ - b² and find its value when a = -7 and b = -3. So, buckle up, and let's get started!

Understanding Algebraic Expressions

Before we jump into solving the problem, let's make sure we're all on the same page about what algebraic expressions are. An algebraic expression is a combination of variables (like a and b), constants (numbers), and mathematical operations (like addition, subtraction, multiplication, and exponentiation). In our case, the expression a³ - b² involves the variables a and b, exponents, and subtraction.

The key to evaluating an algebraic expression is substitution. This means replacing the variables with their given values. Once we've substituted, we follow the order of operations (PEMDAS/BODMAS) to simplify the expression and arrive at our final answer.

Defining Variables and Constants

In algebraic expressions, variables are symbols (usually letters) that represent unknown or changing values. In our problem, a and b are the variables. Constants, on the other hand, are fixed numerical values. When we're given values for the variables, like a = -7 and b = -3, we can substitute these constants into the expression.

The Order of Operations (PEMDAS/BODMAS)

To ensure we evaluate expressions correctly, we must follow the order of operations. This is often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). These acronyms tell us the order in which we should perform operations:

  1. Parentheses / Brackets: Evaluate expressions inside parentheses or brackets first.
  2. Exponents / Orders: Calculate exponents (powers) and roots.
  3. Multiplication and Division: Perform multiplication and division from left to right.
  4. Addition and Subtraction: Perform addition and subtraction from left to right.

Solving the Expression a³ - b²

Now that we've covered the basics, let's tackle our problem: evaluating a³ - b² when a = -7 and b = -3. We'll break this down into clear, manageable steps.

Step 1: Substitution

The first step is to substitute the given values of a and b into the expression. We replace a with -7 and b with -3:

a³ - b² = (-7)³ - (-3)²

Substitution is a crucial step, so make sure you replace the variables carefully with their corresponding values. Pay close attention to signs (positive and negative) as they can significantly impact the final result.

Step 2: Evaluating Exponents

Next, we need to evaluate the exponents. Remember, an exponent indicates how many times a number is multiplied by itself. So, (-7)³ means -7 multiplied by itself three times, and (-3)² means -3 multiplied by itself twice.

Let's calculate these:

  • (-7)³ = (-7) * (-7) * (-7) = -343
  • (-3)² = (-3) * (-3) = 9

Now, we substitute these values back into our expression:

(-7)³ - (-3)² = -343 - 9

Step 3: Performing Subtraction

The final step is to perform the subtraction. We have -343 - 9. Subtracting a positive number is the same as adding a negative number, so this is the same as -343 + (-9).

Combining these, we get:

-343 - 9 = -352

Therefore, the value of the expression a³ - b² when a = -7 and b = -3 is -352.

Common Mistakes to Avoid

Evaluating algebraic expressions can sometimes be tricky, and it's easy to make mistakes if you're not careful. Here are a few common pitfalls to watch out for:

Sign Errors

One of the most frequent mistakes is getting the signs wrong, especially when dealing with negative numbers. Remember that a negative number raised to an odd power is negative, while a negative number raised to an even power is positive. For example, (-2)³ = -8, but (-2)² = 4. Always double-check your signs to avoid errors.

Order of Operations

Failing to follow the order of operations (PEMDAS/BODMAS) is another common mistake. Make sure you evaluate exponents before multiplication and division, and multiplication and division before addition and subtraction. If you're unsure, write out each step clearly to help you stay organized.

Misinterpreting Exponents

It's crucial to understand what exponents mean. A common mistake is to multiply the base by the exponent instead of raising it to the power. For example, 2³ is 2 * 2 * 2 = 8, not 2 * 3 = 6. Be sure to calculate the power correctly.

Careless Arithmetic

Even if you understand the concepts, simple arithmetic errors can lead to the wrong answer. Take your time, double-check your calculations, and use a calculator if needed, especially for larger numbers. It’s always better to be accurate than to rush and make a mistake.

Practice Problems

To solidify your understanding, let's try a few more practice problems. Evaluating algebraic expressions is a skill that improves with practice, so work through these problems carefully.

Practice Problem 1

Evaluate the expression 2x² + 3y when x = 4 and y = -2.

  1. Substitute:

    2(4)² + 3(-2)

  2. Exponents:

    2(16) + 3(-2)

  3. Multiplication:

    32 + (-6)

  4. Addition:

    32 - 6 = 26

So, the value of the expression is 26.

Practice Problem 2

Evaluate the expression (p - q)² when p = -5 and q = 3.

  1. Substitute:

    (-5 - 3)²

  2. Parentheses:

    (-8)²

  3. Exponents:

    64

Therefore, the value of the expression is 64.

Practice Problem 3

Evaluate the expression m³ - 4n when m = -2 and n = 5.

  1. Substitute:

    (-2)³ - 4(5)

  2. Exponents:

    -8 - 4(5)

  3. Multiplication:

    -8 - 20

  4. Subtraction:

    -28

So, the value of the expression is -28.

Conclusion Evaluating Algebraic Expressions

Great job, guys! You've now learned how to evaluate algebraic expressions by substituting values for variables and following the order of operations. We walked through a specific example, discussed common mistakes to avoid, and worked through some practice problems. Remember, the key to mastering algebra is practice, so keep working on problems, and you'll become more confident and proficient.

Algebra is a fundamental part of mathematics and is used in many real-world applications, from engineering and physics to economics and computer science. By understanding how to evaluate expressions, you're building a solid foundation for more advanced mathematical concepts. So, keep up the great work, and happy solving!