Problem 27 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The simplified expression is 21.

Step 1: Simplify Inside Parentheses

Identify and simplify the expression inside the parentheses. For the expression \(5 - 2\), perform the subtraction to simplify: \(5 - 2 = 3\). This changes the expression to \(2 + 3 + 4^2\).

Step 2: Address Exponents

Next, evaluate the exponentiation. The expression \(4^2\) stands for \(4 \times 4\), which equals \(16\). The resulting expression becomes \(2 + 3 + 16\).

Step 3: Perform Addition

Now, add the remaining terms from left to right. First, add \(2 + 3\), which equals \(5\). Next, add \(5 + 16\), which equals \(21\). Thus, the simplified expression is \(21\).

Key Concepts

Order of Operations

In mathematics, the **Order of Operations** is a set of rules that determine the sequence in which calculations should be performed to ensure accurate results. The standard order is often remembered by the acronym PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

When simplifying expressions, always begin with calculations inside parentheses, as these operations override all others. This first step ensures any immediate calculations within the parentheses are completed before moving on to exponents and other operations. By strictly following this sequence, any mathematical expression can be simplified accurately.

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. In an expression like \(4^2\), the base 4 is multiplied by itself, and the 2 is the exponent telling us how many times to perform the multiplication:

• 4 times 4, which equals 16.

Exponents are calculated after any operations inside parentheses had been completed but before performing any multiplication, division, addition, or subtraction. Understanding exponents enables more efficient handling of large numbers and simplifies computations significantly. The quick simplification process of \(4^2\) in our example shows how powerful this operation is.

Integers

**Integers** are a type of number that includes all whole numbers, both positive and negative, as well as zero. These numbers are fundamental to arithmetic operations and make up the primary unit in various mathematical contexts.

• Positive integers: 1, 2, 3, ...
• Negative integers: -1, -2, -3, ...
• Zero, which acts as the neutral element in addition and subtraction

The expression involving integers often remains whole-numbered at each step, as seen in our simplification process. Handling integers doesn't involve fractions or decimals, which simplifies arithmetic operations and maintains calculations as straightforward and manageable, essential for beginners.

Arithmetic Operations

Arithmetic operations are the basics of mathematics, consisting of addition, subtraction, multiplication, and division. In the given expression and solution, focus primarily on:

• Addition: combining numbers to obtain a total.
• Subtraction: removing a number from another, finding the difference.

The expression \(2 + (5-2) + 4^2\) involves both addition and subtraction. Initially, subtraction calculates the inside parentheses value. As the sequence progresses, addition comes into play, linking simplified terms into the final sum. Mastering these basic operations facilitates more complicated mathematical concepts and techniques. Every mathematical equation or expression simplifies according to these fundamental rules.