Problem 25 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(2+(5-2)+4^{2}\)

Step-by-Step Solution

Answer

The simplified expression is 21.

Step 1: Evaluate the Parentheses

First, solve the operation inside the parentheses: \(5-2\). This equals \(3\).

Step 2: Replace the Parentheses With the Result

Substitute the value obtained in Step 1 back into the expression, which changes it to \(2 + 3 + 4^2\).

Step 3: Evaluate the Exponent

Calculate the exponent in the expression: \(4^2\), which is equal to \(16\).

Step 4: Replace the Exponent With the Result

Substitute the value obtained in Step 3 back into the expression, which now becomes \(2 + 3 + 16\).

Step 5: Perform Addition

Add the numbers in sequence: \(2 + 3 = 5\) and then add \(5 + 16 = 21\).

Key Concepts

Order of Operations

When simplifying mathematical expressions, it's crucial to follow a specific sequence, known as the order of operations. This process ensures that no matter who performs the calculations, the same result is achieved. The sequence to follow can be easily remembered with the acronym PEMDAS.

• P: Parentheses
• E: Exponents
• M/D: Multiplication/Division (from left to right)
• A/S: Addition/Subtraction (from left to right)

In our example, the expression is given as \(2 + (5-2) + 4^{2}\). We start by solving what's inside the parentheses, \(5-2\). Once the parentheses are resolved, we proceed to exponents, calculating \(4^{2}\). Only after these steps do we move to addition. This sequence minimizes errors and confusion.

Exponents in Algebra

Exponents are a way to express repeated multiplication of the same factor. In algebra, when you see an expression like \(4^2\), it's shorthand for \(4 \times 4\). It's essential to tackle exponents early in the process because they can significantly change the value within an equation.

In our exercise, after the parentheses, we encounter the expression \(4^2\). Solving this, we replace the exponent by its calculated value, which is \(16\). By addressing exponents accurately and in their turn according to the order of operations, we maintain the integrity of our calculations. Remember, not handling exponents at the right time can lead to incorrect final results.

Addition and Subtraction in Algebra

In the final steps of simplifying an expression, we deal with addition and subtraction. These operations need straightforward calculations, but they must be done in the sequence established by the order of operations.

In our example, once the parentheses and exponents are resolved, we are left with adding the numbers: \(2 + 3 + 16\). Start by adding the first two numbers, \(2 + 3 = 5\), and then add the result to the next number: \(5 + 16 = 21\).

• Always add or subtract numbers as they appear from left to right.
• Ensure no brackets or higher precedence operations are pending.

This careful approach helps achieve an accurate simplification of algebraic expressions.