Problem 53 - Beginning and Intermediate Algebra

Question

Simplify each expression. (Remember the order of operations.) See Examples 4 and 5. $$ (3-6)+4^{2} $$

Step-by-Step Solution

Answer

The simplified expression is 13.

Step 1: Evaluate the Exponentiation

According to the order of operations, we need to resolve the exponentiation first. In the expression \(4^2\), calculate the result by multiplying the base by itself: \[4^2 = 4 \times 4 = 16\].

Step 2: Simplify Inside the Parentheses

Next, simplify any expressions inside the parentheses. The expression inside the parentheses is \(3-6\), which we solve as follows:\[3 - 6 = -3\].

Step 3: Combine Results

Now, combine the simplified results from Steps 1 and 2. The expression becomes \[-3 + 16\].

Step 4: Perform the Addition

Perform the addition operation:\[-3 + 16 = 13\].

Key Concepts

Exponentiation

Exponentiation is an operation involving a base and an exponent. The base is multiplied by itself as many times as the exponent suggests. In the expression \(4^2\), the base is 4 and the exponent is 2. This means we calculate \(4 \times 4 = 16\). Exponentiation takes priority over other operations due to the order of operations, often abbreviated as PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This hierarchy helps in solving expressions systematically.

• In our example: \(4^2\) simplifies to 16.
• Role of exponents: Great for expressing repeated multiplications.

Understanding exponentiation is crucial, especially when dealing with complex problems involving powers.

Parentheses

Parentheses are like a signal in mathematics that tell us to focus on the operations within them first, after any exponentiations. When you see parentheses in an expression, always perform these operations before moving on to others outside. In the expression \((3 - 6) + 4^2\), they prompt us to solve the \(3 - 6\) part first, following the evaluation of any exponentiations.

• In our example: Solve inside \((3 - 6)\) to get \(-3\).
• Importance: Ensures clarity and prevents misinterpretation of the order.

By simplifying the inside of the parentheses, we get much closer to the final solution.

Simplification

Simplification in mathematics involves reducing expressions to their simplest form. Through step-by-step evaluations, we refine the expression until it's as straightforward as possible. In this exercise, after evaluating the exponentiation and simplifying the parentheses, we are left with \(-3 + 16\).

• Simplified expression: This takes us from a complex expression to something that is easily solvable.
• Example: \(-3\) comes from the parentheses, and 16 from \(4^2\).

Simplifying expressions not only makes them easier to work with but also helps in identifying the right order of solving the operations per the given problem.

Addition

Addition is one of the basic operations in arithmetic, dealing with combining numbers to produce a sum. After simplifying the expression to \(-3 + 16\), the next step is straightforward addition. Start from the simplified sum, keeping the signs of the numbers in mind, we find:

• Calculation: Begin with \(-3 + 16\), which results in 13.
• Result interpretation: The negative number is subtracted from the positive, as it effectively decreases the sum.

This final step resolves the expression to its simplest, most understandable form, demonstrating the seamless flow from exponentiation to a completed addition.