Simplify: $(-2)^2 - 4(-3)(2$\]

Feb 28, 2025 by ADMIN 31 views

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. It involves combining like terms, removing unnecessary operations, and rearranging the expression to make it easier to work with. In this article, we will simplify the given expression $(-2)^2 - 4(-3)(2)$ using basic algebraic rules and properties.

Understanding the Expression

The given expression is $(-2)^2 - 4(-3)(2)$ . To simplify this expression, we need to follow the order of operations (PEMDAS):

Evaluate the exponentiation: $(-2)^2$
Multiply the numbers: $4(-3)(2)$
Subtract the results of the two operations

Step 1: Evaluate the Exponentiation

The first step is to evaluate the exponentiation: $(-2)^2$ . According to the exponentiation rule, when a negative number is raised to an even power, the result is positive.

$(-2)^2 = (-2) \times (-2) = 4$

So, the expression becomes: $4 - 4(-3)(2)$

Step 2: Multiply the Numbers

The next step is to multiply the numbers: $4(-3)(2)$ . We need to follow the order of operations and multiply the numbers from left to right.

$4(-3)(2) = 4 \times (-3) \times 2 = -24$

So, the expression becomes: $4 - (-24)$

Step 3: Simplify the Expression

Now, we need to simplify the expression: $4 - (-24)$ . To simplify this expression, we need to remove the negative sign from the second term.

$4 - (-24) = 4 + 24 = 28$

Therefore, the simplified expression is: $28$

Conclusion

In this article, we simplified the given expression $(-2)^2 - 4(-3)(2)$ using basic algebraic rules and properties. We followed the order of operations (PEMDAS) and evaluated the exponentiation, multiplied the numbers, and simplified the expression to get the final result. This example demonstrates the importance of simplifying expressions in mathematics and how it can help us solve problems efficiently.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions:

Follow the order of operations (PEMDAS)
Evaluate exponentiation first
Multiply numbers from left to right
Remove negative signs from terms
Combine like terms

By following these tips and tricks, you can simplify expressions efficiently and accurately.

Common Mistakes

Here are some common mistakes to avoid when simplifying expressions:

Not following the order of operations (PEMDAS)
Evaluating expressions from left to right instead of following the order of operations
Not removing negative signs from terms
Not combining like terms

By avoiding these common mistakes, you can simplify expressions accurately and efficiently.

Real-World Applications

Simplifying expressions has many real-world applications in mathematics and other fields. Here are a few examples:

Algebra: Simplifying expressions is a crucial skill in algebra, where we use variables and constants to solve equations and inequalities.
Calculus: Simplifying expressions is also important in calculus, where we use limits, derivatives, and integrals to solve problems.
Physics: Simplifying expressions is essential in physics, where we use mathematical models to describe the behavior of physical systems.
Engineering: Simplifying expressions is also important in engineering, where we use mathematical models to design and optimize systems.

By simplifying expressions, we can solve problems efficiently and accurately, which is essential in many real-world applications.

Final Thoughts

Introduction

In our previous article, we simplified the expression $(-2)^2 - 4(-3)(2)$ using basic algebraic rules and properties. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. PEMDAS stands for:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next (e.g., 2^3).
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify expressions with negative numbers?

A: When simplifying expressions with negative numbers, remember to follow the order of operations (PEMDAS). If you have a negative number raised to an even power, the result is positive. For example:

$(-2)^2 = 4$

If you have a negative number raised to an odd power, the result is negative. For example:

$(-2)^3 = -8$

Q: How do I simplify expressions with fractions?

A: When simplifying expressions with fractions, remember to follow the order of operations (PEMDAS). If you have a fraction with a negative sign, you can simplify it by multiplying the numerator and denominator by -1. For example:

$-\frac{1}{2} = \frac{-1}{2}$

Q: How do I simplify expressions with variables?

A: When simplifying expressions with variables, remember to follow the order of operations (PEMDAS). If you have a variable raised to an even power, the result is positive. For example:

$x^2 = x \times x = x^2$

If you have a variable raised to an odd power, the result is negative. For example:

$-x^3 = -x \times x \times x = -x^3$

Q: What are some common mistakes to avoid when simplifying expressions?

A: Here are some common mistakes to avoid when simplifying expressions:

Not following the order of operations (PEMDAS)
Evaluating expressions from left to right instead of following the order of operations
Not removing negative signs from terms
Not combining like terms

Q: How do I simplify expressions with multiple operations?

A: When simplifying expressions with multiple operations, remember to follow the order of operations (PEMDAS). Evaluate any parentheses first, then any exponents, followed by any multiplication and division operations, and finally any addition and subtraction operations.

Q: Can I use a calculator to simplify expressions?

A: Yes, you can use a calculator to simplify expressions. However, it's always a good idea to double-check your work by simplifying the expression manually.