Problem 47 - Introductory Algebra for College Students

Question

In Exercises \(29-72,\) use the order of operations to simplify each expression. $$[2(6-2)]^{2}$$

Step-by-Step Solution

Answer

The simplified version of the expression \( [2(6-2)]^{2} \) is 64.

Step 1: Simplify within parentheses

The first operation according to the order of operations is parentheses. In the expression \( [2(6-2)]^{2} \), this means performing the subtraction within the parentheses. So, \( 6-2=4 \). Thus, the expression simplifies to \( [2(4)]^{2} \).

Step 2: Perform multiplication

Next, multiplication is performed. So, the multiplication of 2 and 4 gives 8. Thus, the expression simplifies to \( 8^{2} \).

Step 3: Raise to the power

Finally, the given expression is simplified by applying the exponent \( 2 \) to the number \( 8 \). The square of 8 is 64.

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking down a mathematical equation into a simpler form through a series of logical steps. In the problem discussed, the expression given is \(\ [2(6-2)]^2 \). The goal of simplifying is to reach the most basic numerical answer. This process necessitates the use of the order of operations, a set of rules that prioritize different mathematical processes.

• Begin by examining each component part of the expression. This often involves handling operations within parentheses first.
• Step by step, each operation is dealt with according to these priorities until the expression collapses into a single number.

Breaking it down step-by-step helps avoid errors and ensures everyone will get the same result. It makes complex expressions easier to understand and manage, and it helps focus on completing one task at a time.

Mathematical Operations

Mathematical operations consist of addition, subtraction, multiplication, and division, as well as more complex operations such as exponents and roots. The order in which these operations are executed is crucial and is governed by the PEMDAS/BODMAS rule: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

• The initial step in our exercise is dealing with what is inside the parentheses \(\ (6-2) \), which simplifies to 4.
• Following this, multiplication is executed \(\ 2 \times 4 = 8 \).
• Finally, apply the exponent to this new result.

This order prevents chaos and ensures that everyone solves mathematical expressions in the same way, maintaining consistency and accuracy in mathematics.

Exponents

Exponents are a specific type of mathematical operation where a number is multiplied by itself a specific number of times, indicated by the exponent. In this scenario, the base number 8 is raised to the power of 2. \

• The term \(\ 8^2 \) means multiplying 8 by itself, resulting in \(\ 8 \times 8 = 64 \).
• This operation is visually represented by a small raised number, the exponent, next to the base number.
• Exponents simplify the repeated multiplication process, making expressions more concise.

By applying the exponent, you effectively finish simplifying the given expression to a single numerical answer, which is straightforward and easy to interpret.