Problem 93 - Prealgebra

Question

The problems below review some of the material on solving equations. Reviewing these problems will help you with the next section. Solve. $$3^{2}+4^{2}$$

Step-by-Step Solution

Answer

25

Step 1: Understand the Problem

The problem asks us to evaluate the expression \(3^2 + 4^2\). This involves finding the squares of both 3 and 4 and then adding the results together.

Step 2: Calculate the Squares

First, calculate the square of 3. \(3^2 = 3 \times 3 = 9\). Next, calculate the square of 4. \(4^2 = 4 \times 4 = 16\).

Step 3: Add the Results

Now, add the results of the squared numbers. \(9 + 16 = 25\).

Key Concepts

Solving Equations

In prealgebra, solving equations is a fundamental skill that involves finding the value of an unknown that makes an equation true.

Although the exercise provided doesn't directly involve solving for an unknown, understanding how to solve basic mathematical equations is crucial for handling more complex problems in algebra.

To solve an equation, you perform operations to isolate the unknown variable on one side of the equation. For example:

• If the equation is simple, like \(x + 5 = 10\), you'd subtract 5 from both sides to find \(x = 5\).
• For equations involving multiplication, like \(3x = 12\), you'd divide both sides by 3, leading to \(x = 4\).

These basic operations, learned and practiced with numbers, prepare you for more complex algebraic equations.

Each step in solving an equation requires precision and care to ensure that the integrity of the equation remains intact.

Squaring Numbers

Squaring numbers is an essential concept in mathematics, particularly in prealgebra.

When you square a number, you multiply it by itself. This operation is denoted by an exponent of 2, located as a superscript to the right of the base number.

Let's break down the squaring process:

• For the number 3, squaring (\(3^2\)) means calculating \(3 \times 3\), which equals 9.
• Similarly, for the number 4, squaring (\(4^2\)) means \(4 \times 4\), resulting in 16.

Understanding squaring is crucial because it appears frequently in different areas of mathematics such as quadratic equations, geometry, and engineering.

Being comfortable with squaring numerals prepares you for these more advanced applications. Practice by squaring small integers until you can do it with ease.

Mathematical Operations

Mathematical operations form the backbone of solving equations in prealgebra. These include addition, subtraction, multiplication, and division.

Each operation plays a vital role in solving equations and completing exercises like the one provided, involving the evaluation of expressions.

• Addition: Used in our exercise when the two squared numbers, 9 and 16, are combined to make 25.
• Subtraction: Though not directly used in this exercise, it helps in solving equations by removing or cancelling terms.
• Multiplication: Critical in understanding squaring, as seen when \(3 \times 3\) and \(4 \times 4\) are calculated.
• Division: Used in isolating variables during the solving of equations, often introduced after basic operations are understood.

Mastering these operations enables fluid movement through more complex algebraic manipulations. Practice each in isolation and then in combination, as in the provided exercise.