Problem 46 - Algebra A Combined Function

Question

Simplify each expression. $$ 3+4[8(5 \cdot 5-20)-41] $$

Step-by-Step Solution

Answer

The expression simplifies to -1.

Step 1: Simplify Inside the Parentheses

Start by addressing the innermost part of the expression, which is inside the parentheses: \(5 \cdot 5 - 20\). First, calculate \(5 \cdot 5 = 25\).Then, subtract 20 from 25: \(25 - 20 = 5\). Now the expression updates to: \[3 + 4[8(5) - 41]\]

Step 2: Simplify Inside the Brackets

After simplifying the parentheses, move to the brackets: \(8(5)\).Calculate \(8 \cdot 5 = 40\).Now, substitute back into the expression: \[3 + 4[40 - 41]\].

Step 3: Simplify the Subtraction

Inside the brackets, perform the subtraction \(40 - 41\):This results in \(-1\).So the expression now is: \[3 + 4(-1)\].

Step 4: Multiply Inside the Brackets

Perform the multiplication inside the brackets: \(4(-1)\).This yields \(-4\).Update the expression to: \[3 - 4\].

Step 5: Simplify the Final Expression

Finally, perform the subtraction: \(3 - 4\).This results in \(-1\).

Key Concepts

Order of Operations

The order of operations is a fundamental principle in algebra that ensures expressions are simplified consistently and correctly. It's commonly remembered by the acronym PEMDAS:

• **P**: Parentheses first
• **E**: Exponents (i.e., powers and square roots, etc.)
• **MD**: Multiplication and Division (left-to-right)
• **AS**: Addition and Subtraction (left-to-right)

In our example, following PEMDAS correctly is crucial. We begin by simplifying inside the parentheses, then move on to handle operations inside the brackets. After that, multiplication and subtraction across the expression are carried out in sequence. This ensures each step follows a logical progression, preventing errors in calculation.

Parentheses and Brackets

Parentheses and brackets help organize mathematical expressions and dictate the order of operations within them. They act as signals to perform operations enclosed within them before addressing other parts of the expression.
In our original exercise, we encounter both parentheses and brackets. By focusing first on the contents within the innermost parentheses \(5 \cdot 5 - 20\), and then proceeding to the brackets, \[8(5) - 41\], this ensures that all internal operations are executed correctly and accurately before moving to the rest of the expression. This ordered approach significantly influences the final outcome of the evaluation.

Expression Evaluation

Expression evaluation is the process of calculating the value of an algebraic expression. It involves following the correct order of operations, as well as dealing properly with parentheses and brackets.

Simplifying the Example Step-by-Step

Starting with the expression \(3 + 4[8(5 \cdot 5 - 20) - 41]\), we first simplify the inner expression to \(5\) by performing the multiplication and subtraction. Next, calculate \(8 \cdot 5\) which simplifies to \(40\), and then perform the subtraction to get \([-1]\).
This leads to further simplification where we multiply \(4 \cdot -1\) to reach \(-4\). Finally, the expression evaluates to a clean \(-1\) after the last step of subtraction, completing the process of evaluation. Understanding each stage of simplification helps reveal the intricate structure and balance within mathematical expressions.