Problem 26 - Elementary Algebra

Question

For the following problems, use the order of operations to find each value. $$2\\{(7+7)+6[4(8+2)]\\}$$

Step-by-Step Solution

Answer

Answer: The value of the expression is 508.

Step 1: Simplify the inner parentheses

First, we need to simplify the expression inside the inner parentheses. This is \((8+2)\), which equals \(10\). The expression now becomes: $$2\\{(7+7)+6[4(10)]\\}$$

Step 2: Simplify the expression within square brackets

Now, we will deal with the expression within the square brackets. This includes the multiplication \(4(10)\), which equals \(40\). The expression now becomes: $$2\\{(7+7)+6[40]\\}$$

Step 3: Simplify the expression within curly brackets

We will now simplify the remaining expression inside the curly brackets: \((7+7)\) and \(6[40]\). These are addition and multiplication operations, which will be done from left to right: \((7+7)=14\) \(6[40]=240\) Now the expression looks like this: $$2\\{(14)+240\\}$$

Step 4: Simplify the entire expression

Finally, all we need to do is to simplify the remaining expression, which involves a multiplication operation: \(2\\{(14)+240\\}=2(14+240)=2(254)=508\) Thus, the value of the expression is 508.

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (like add, subtract, multiply, and divide). To evaluate an algebraic expression, one must combine like terms and follow a specific order of operations, often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Our exercise involves an algebraic expression with multiple layers of operations, including parentheses, brackets, and curly braces. Each layer has to be simplified according to the order of operations, starting with the innermost parentheses and working outward. Understanding algebraic expressions is fundamental in algebra and vital for solving equations and inequalities. Correctly evaluating these expressions requires a step-by-step approach to ensure that no part of the expression is overlooked and all operations are executed in the correct sequence.

Parentheses Simplification

The process of parentheses simplification is crucial when dealing with complex algebraic expressions. It involves solving anything within parentheses as a priority before moving on to other operations. Nested parentheses, brackets, or braces require you to start from the innermost level and work your way out, simplifying each level one step at a time.

In our problem, we've encountered both parentheses and brackets. The importance of simplifying the parentheses before proceeding is highlighted in this step-by-step solution: we first tackle \(8+2\), the innermost parentheses, before proceeding to the multiplication and addition that follow. Failure to simplify the innermost parentheses first can lead to a wrong answer, which emphasizes the need to follow the order of operations strictly. By tackling each operation in the right sequence, we ensure the algebraic expression is simplified correctly.

Multiplication

Among the basic arithmetic operations, multiplication is a shortcut for repeated addition. It is a crucial operation in algebra and takes precedence over addition and subtraction but comes after any parentheses or exponents have been dealt with. Multiplication in algebraic expressions can appear in different ways: as a cross \(x\), a dot \(\cdot\), or even implied by placing parentheses or numbers next to each other, as seen with \(6[40]\) in our problem.

Correct application of multiplication is demonstrated in the solution by multiplying \(4\times 10\) inside the brackets, and then later multiplying \(2(14 + 240)\) as the final step. Remember that if any term is adjacent to parentheses, it signifies multiplication with the contents of the parentheses. On encountering such situations, it's important to resolve the multiplication only after the expressions within any kind of groupings (like parentheses or brackets) have been simplified. This follows the hierarchy of the order of operations and makes sure that we arrive at the correct final solution.