Problem 43 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$(3+4)^{2}$$

Step-by-Step Solution

Answer

The expression simplifies to 49.

Step 1: Calculate Inside the Parentheses

First, solve the expression inside the parentheses: \(3 + 4\). When you add these numbers together, you get \(7\).

Step 2: Raise the Result to the Power of 2

Now take the result from Step 1, which is \(7\), and calculate \(7^2\) by multiplying 7 by itself: \(7 \times 7 = 49\).

Key Concepts

Order of Operations

Understanding the order of operations is crucial in mathematics. It dictates the sequence in which operations should be performed to ensure consistency and accuracy in solving expressions. The standard rule to remember is "PEMDAS":

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

This sequence ensures that everyone simplifies expressions in the same way. For example, in the expression (3 + 4)^{2}, parentheses are handled first due to their priority, followed by exponents.

Exponents

Exponents are a mathematical operation where a number, known as the base, is multiplied by itself a specific number of times, indicated by the exponent. It is expressed in the form of \(a^b\), where \(a\) is the base and \(b\) is the exponent. In our exercise (3 + 4)^{2}, after solving the operation inside the parentheses, you have 7^{2}. This means 7 is multiplied by itself: \(7 \times 7\), resulting in 49. Exponents dramatically increase the value of numbers with just small changes in the base or exponent.

Addition

Addition is one of the basic operations of arithmetic, used to combine two or more numbers into a single sum. In this numerical expression, 3 + 4, addition is performed first as it is enclosed within parentheses. When you add 3 and 4, it results in 7. Understanding addition is fundamental, as it is the building block for more complex operations like multiplication or working within parentheses, as seen in this exercise.

Parentheses

Parentheses in mathematics dictate which parts of an expression are calculated first. Using parentheses, as in (3 + 4)^{2}, separates specific operations from others based on priority. This helps in clarifying the order of operations, ensuring that what's enclosed in the parentheses is addressed before moving on to other calculations like exponents or multiplication. Parentheses are crucial for determining the correct result, especially in more intricate expressions.