Problem 87 - Algebra A Combined Function

Question

Evaluate each expression. \(2^{2}-3[(2-8)-(-6-8)]\)

Step-by-Step Solution

Answer

The value of the expression is -20.

Step 1: Evaluate Inside the Parentheses

The expression inside the parentheses is \((2-8)-(-6-8)\). First solve \(2-8\) which equals \(-6\), and then solve \((-6-8)\), which equals \(-14\).

Step 2: Simplify Inside the Brackets

Now simplify inside the brackets: the expression \(-6 - (-14)\) becomes \(-6 + 14\) after distributing the negative sign, which simplifies to \(8\).

Step 3: Evaluate the Exponent

Now move outside the brackets to evaluate the exponent: Calculate \(2^2 = 4\).

Step 4: Substitute Values and Final Calculation

Substitute the evaluations back into the expression: The expression becomes \(4 - 3[8]\). Evaluate \(3[8] = 24\). Finally, calculate \(4 - 24 = -20\).

Key Concepts

Evaluating Expressions

In mathematics, evaluating expressions involves finding the value of a given expression using the correct order of operations.
Understanding and applying the appropriate steps can simplify complex expressions into manageable calculations.

• First, identify all the different parts of the expression, including coefficients, variables, constants, and operations.
• Work through calculations step-by-step, usually starting from the innermost part of an expression and working outward.
• Use arithmetic operations, such as addition, subtraction, multiplication, and division, guided by prioritization rules, often referenced as PEMDAS—parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right).

Evaluating expressions requires patience and precision as missteps at any stage can lead to incorrect results.
Each component must be properly calculated before moving on to the next for the accurate final result.

Parentheses and Brackets

Parentheses ( ) and brackets [ ] are essential in mathematical expressions since they indicate which operations should be performed first. This hierarchy determines the sequence of calculations.
They allow you to organize operations, particularly when expressions include multiple steps.
When you encounter parentheses or brackets in an expression:

• Always perform the operations inside parentheses first, following the order of operations within them.
• Once you've simplified the parentheses, evaluate any brackets by treating them similarly as you would parentheses.
• Be mindful of nested parentheses or brackets: solve the innermost pair first and work outward.

Using parentheses and brackets correctly helps prevent errors and ensures that expressions reflect the intended mathematical operations clearly.
This ultimately allows for the correct calculation of the result.

Exponents

Exponents represent repeated multiplication of a number by itself. An exponent is shown as a small number to the upper-right of a base number. For instance, in the expression\(2^2\), the "2" is the base and it is raised to the power of "2."
To understand exponents in expressions:

• Recognize that multiplying the base number by itself as many times as indicated by the exponent. For\(2^2\), you multiply 2 × 2.
• Apply the correct operations, such as calculating exponents just after solving what’s in the parentheses but before any multiplication or division that lies outside those parentheses.

Exponents fundamentally change the value of the base number, often resulting in much larger numbers.
Grasping how to compute and when to apply them is crucial in evaluating complex mathematical expressions.