Problem 80 - Algebra 1: Concepts and Skills

Question

NUMERICAL EXPRESSIONS Evaluate the expression. Then simplify the answer. (Lesson 1.3) $$ (28 \div 4)+3^{2} $$

Step-by-Step Solution

Answer

The simplified version of the expression \( (28 \div 4)+3^{2} \) is 16.

Step 1: Perform division operation

Since division comes before addition in the order of operations, do the division first. Divide 28 by 4 to get 7. So the expression becomes \( 7+3^{2} \).

Step 2: Handle the exponent

Next, according to the order of operations, is handling any exponents. In this case, we have \( 3^{2} \). Squaring a number means multiplying the number by itself. So, \( 3^{2} = 3 * 3 \), which equals 9. Therefore, the expression now becomes \( 7+9 \).

Step 3: Perform addition operation

The last operation left to perform is addition. Add 7 and 9 to get 16. Therefore, the final simplified expression is 16.

Key Concepts

Order of Operations

When it comes to evaluating numerical expressions, it is crucial to follow the order of operations, a fundamental guideline ensuring consistent and accurate results.
Think of the order of operations as a rulebook for tackling math problems: it tells you which calculations to perform first to end up with the correct answer. This rulebook is often memorized using the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
In the case of the expression \[\begin{equation}(28 \div 4)+3^{2}\end{equation}\], you first address any calculations inside parentheses, then handle exponents, followed by multiplication and division in sequence from left to right, and finally perform addition and subtraction, also from left to right. By strictly adhering to PEMDAS, mathematical expressions are greatly simplified.

Simplifying Expressions

Simplifying expressions is about making complex, or cluttered, mathematical sentences easier to understand and solve. It encourages clarity and minimizes the chance of making errors.
To simplify an expression, identify and perform operations systematically based on the order of operations. Start by eliminating parentheses through performing the operations inside them, handle exponents, then perform multiplications and divisions, and end with additions and subtractions. Simplifying helps to transform an unmanageable expression into a more digestible and easily calculable form. For instance, in the expression \[\begin{equation}(28 \div 4)+3^{2}\end{equation}\], simplifying involves executing division first, followed by exponentiation, and finally wrapping up with addition, ultimately giving us a simple number: 16.

Exponents and Powers

Exponents, also known as powers, are shorthand for repeated multiplication. They play a critical role in simplifying expressions that contain repeated multiplication of the same number.
The exponent \[\begin{equation}3^{2}\end{equation}\], for instance, tells you to multiply 3 by itself, resulting in \[\begin{equation}3 \times 3 = 9\end{equation}\]. Understanding how to work with exponents is vital as they are second in line after parentheses in the order of operations. A solid grasp of exponents allows for quick simplification of parts of larger numerical expressions, making them less intimidating and easier to evaluate.