Problem 50 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the order of operations to simplify each expression. $$3[4+3(10-8)]$$

Step-by-Step Solution

Answer

The simplified expression is 30.

Step 1: Solve the Parentheses

According to the order of operations rule, first calculate what is inside the parentheses: \(3[4+3(10-8)] = 3[4+3(2)]\).

Step 2: Perform Multiplication

Secondly, calculate the multiplication operation: \(3[4+3(2)] = 3[4+6]\).

Step 3: Perform addition operation

Now, perform the operation in brackets by adding the numbers: \(3[4+6] = 3[10]\).

Step 4: Perform Multiplication

Finally, perform multiplication: \(3[10] = 30\).

Key Concepts

Simplifying Expressions

Understanding how to simplify expressions is a fundamental skill in algebra. Simplification involves reducing expressions to their most basic form while maintaining their original value. It's essential to follow the standard order of operations, known as PEMDAS or BODMAS, when simplifying.

This mnemonic stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). To master simplifying expressions, practice is key. Start with substituting numbers into the expression to check your work, and gradually, you'll be able to solve more complex problems with ease.

With the given exercise, for instance, each step we take aligns closely with this order: addressing operations within the parentheses, then multiplying, and finally adding terms before multiplying one last time to find the simplified form.

Parentheses in Algebra

The use of parentheses in algebra serves more than just as a organizational tool; they dictate the priority of operations. When an expression contains parentheses, it's a signal to address these segments first before proceeding to other calculations. This is why in our example exercise, we focused on the content within the parentheses in the first step.

Keep in mind that parentheses can also be nested within each other, creating layers of priorities which must be addressed from the innermost set to the outermost. As you grow comfortable with these, you'll start to see how algebraic expressions are structured in ways that guide you through the process of solving them. Furthermore, parentheses are also used to group terms that are affected by a coefficient, as we see with the final multiplication step in the given problem.

Intermediate Algebra

Intermediate algebra is a stepping stone between basic algebra and more advanced levels of mathematics. It encompasses a wide range of topics, including the manipulation and simplification of algebraic expressions, solving equations and inequalities, and understanding functions. Mastery of intermediate algebra is evident when a student can approach a problem like the one provided, knowing not just the steps, but the rationale behind them.

Intermediate algebra is also where concepts like factoring, rational expressions, and radicals are introduced. These concepts need a strong foundation in the order of operations and an understanding of the properties of numbers to be solved correctly, as they are more intricate and require a deeper level of thought. The exercise provided is a straightforward example illustrating these fundamental principles at play in intermediate algebra.