Simplify The Expression:$7 \cdot (-3) + [2 + 3(-5)]

Mar 12, 2025 by ADMIN 52 views

Simplify the Expression: A Step-by-Step Guide to Evaluating Algebraic Expressions

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will focus on simplifying the expression $7 \cdot (-3) + [2 + 3(-5)]$ . We will break down the expression into smaller parts, apply the order of operations, and use algebraic properties to simplify it.

Understanding the Expression

The given expression is $7 \cdot (-3) + [2 + 3(-5)]$ . To simplify this expression, we need to follow the order of operations, which is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This is often remembered using the acronym PEMDAS.

Step 1: Evaluate the Expression Inside the Parentheses

The expression inside the parentheses is $2 + 3(-5)$ . To evaluate this expression, we need to follow the order of operations. First, we need to multiply 3 and -5, which gives us $-15$ . Then, we add 2 to -15, which gives us $-13$ .

Step 2: Multiply 7 and -3

Next, we need to multiply 7 and -3. Multiplication is a commutative operation, which means that the order of the factors does not change the result. Therefore, $7 \cdot (-3) = -21$ .

Step 3: Add -21 and -13

Finally, we need to add -21 and -13. When we add two negative numbers, we need to add their absolute values and keep the negative sign. Therefore, $-21 + (-13) = -34$ .

Conclusion

In conclusion, the simplified expression is $-34$ . We followed the order of operations, evaluated the expression inside the parentheses, multiplied 7 and -3, and added -21 and -13 to arrive at the final answer.

Tips and Tricks

When simplifying algebraic expressions, it is essential to follow the order of operations.
Use parentheses to group numbers and variables that need to be evaluated first.
Apply algebraic properties, such as the commutative and associative properties of addition and multiplication, to simplify expressions.
Use variables to represent unknown values and constants to represent known values.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications. For example, in physics, algebraic expressions are used to describe the motion of objects. In economics, algebraic expressions are used to model the behavior of markets. In computer science, algebraic expressions are used to write algorithms and programs.

Common Mistakes to Avoid

When simplifying algebraic expressions, there are several common mistakes to avoid. These include:

Failing to follow the order of operations.
Not using parentheses to group numbers and variables that need to be evaluated first.
Not applying algebraic properties, such as the commutative and associative properties of addition and multiplication.
Not using variables to represent unknown values and constants to represent known values.

Final Thoughts

Simplifying algebraic expressions is a crucial skill to master in mathematics. By following the order of operations, evaluating expressions inside parentheses, multiplying numbers, and adding numbers, we can simplify even the most complex expressions. Remember to use variables to represent unknown values and constants to represent known values, and apply algebraic properties to simplify expressions. With practice and patience, you will become proficient in simplifying algebraic expressions and be able to tackle even the most challenging problems.

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Introduction

In our previous article, we explored the concept of simplifying algebraic expressions and walked through a step-by-step guide to evaluating the expression $7 \cdot (-3) + [2 + 3(-5)]$ . In this article, we will address some of the most frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is often remembered using the acronym PEMDAS, which stands for:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
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 evaluate expressions inside parentheses?

A: To evaluate expressions inside parentheses, we need to follow the order of operations. First, we need to evaluate any exponential expressions, then any multiplication and division operations, and finally any addition and subtraction operations.

Q: What is the difference between multiplication and addition?

A: Multiplication and addition are two different operations that have different properties. Multiplication is a commutative operation, which means that the order of the factors does not change the result. For example, $3 \cdot 4 = 4 \cdot 3$ . Addition is also a commutative operation, but it is not associative, which means that the order in which we add numbers can change the result. For example, $(3 + 4) + 5 \neq 3 + (4 + 5)$ .

Q: How do I simplify expressions with variables?

A: To simplify expressions with variables, we need to follow the same rules as we do for numerical expressions. We need to evaluate any exponential expressions, then any multiplication and division operations, and finally any addition and subtraction operations.

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change. A constant is a value that does not change. For example, $x$ is a variable, while $5$ is a constant.

Q: How do I simplify expressions with fractions?

A: To simplify expressions with fractions, we need to follow the same rules as we do for numerical expressions. We need to evaluate any exponential expressions, then any multiplication and division operations, and finally any addition and subtraction operations.

Q: What is the difference between a fraction and a decimal?

A: A fraction is a way of expressing a part of a whole as a ratio of two numbers. A decimal is a way of expressing a fraction as a number with a point. For example, $\frac{1}{2}$ is a fraction, while $0.5$ is a decimal.

Tips and Tricks

When simplifying algebraic expressions, it is essential to follow the order of operations.
Use parentheses to group numbers and variables that need to be evaluated first.
Apply algebraic properties, such as the commutative and associative properties of addition and multiplication, to simplify expressions.
Use variables to represent unknown values and constants to represent known values.

Real-World Applications

Common Mistakes to Avoid

When simplifying algebraic expressions, there are several common mistakes to avoid. These include:

Failing to follow the order of operations.
Not using parentheses to group numbers and variables that need to be evaluated first.
Not applying algebraic properties, such as the commutative and associative properties of addition and multiplication.
Not using variables to represent unknown values and constants to represent known values.