Problem 94 - Prealgebra

Question

The problems below review some of the material on solving equations. Reviewing these problems will help you with the next section. Solve. $$5^{2}+12^{2}$$

Step-by-Step Solution

Answer

The result of the expression is 169.

Step 1: Identify the Expression

The expression given is: \(5^2 + 12^2\). We need to compute the sum of these two squares.

Step 2: Calculate Each Square

First, calculate \(5^2\):\[5^2 = 5 \times 5 = 25\]Next, calculate \(12^2\):\[12^2 = 12 \times 12 = 144\]

Step 3: Add the Squares Together

Now, add the results from Step 2 together: \(5^2 + 12^2 = 25 + 144\).

Step 4: Simplify the Sum

Add the two numbers from the previous step: \[25 + 144 = 169\].

Key Concepts

Exponents

Exponents are a mathematical notation that allow you to express a number multiplied by itself a certain number of times in a concise way. When you have a number like 5 raised to the power of 2, which is written as \(5^2\), it means "multiply 5 by itself." The base, which in this case is 5, is the number that gets multiplied, and the exponent, which is 2, tells us how many times to multiply the base. Here’s a simple way to understand it:

• The base is the number you start with, in our exercise the base number is either 5 or 12.
• The exponent is the little number up to the right of the base indicating how many times to use the base in a multiplication.

Working with exponents helps to simplify large mathematical expressions by reducing repetitive multiplication into a straightforward notation. For instance, instead of writing \(5 \times 5\), just write \(5^2\). This can make problem-solving faster and easier. Remember, exponents only impact the base number, and as such, follow specific rules when solving equations involving them.

Squaring Numbers

Squaring a number is a specific case of using exponents where the exponent is 2. The term "square" comes from the geometric concept of creating a square with equal sides. Therefore, squaring a number refers to multiplying it by itself. This process is foundational in solving various mathematical problems.

• For example, squaring the number 5 can be solved as \(5^2 = 5 \times 5 = 25\).
• Similarly, squaring the number 12 follows the same formula: \(12^2 = 12 \times 12 = 144\).

Squaring is important in many areas of math, including algebra, geometry, and more advanced fields like calculus. When solving equations like the one given, understanding how to correctly square numbers is essential. Squares and perfect squares often play a role in deriving formulas, properties of numbers, or simplifying equations.

Step-by-Step Solutions

Step-by-step solutions are a helpful way to break down complex equations into manageable parts. This process allows for a clear understanding of each phase of the problem-solving journey. Let's take a closer look at how this helps in solving our given equation \(5^2 + 12^2\):

• **Step 1: Identify the Expression** - Recognize what needs to be calculated, in this case, two separate squared terms.
• **Step 2: Calculate Each Square** - Solve each term separately for clarity, \(5^2 = 25\) and \(12^2 = 144\).
• **Step 3: Add the Squares Together** - Combine these individual results, thus \(25 + 144\).
• **Step 4: Simplify the Sum** - Finally, simplify the addition, resulting in 169.

Having solutions broken down step-by-step crafts a clearer path through the problem. It peels back layers of complexity and can be particularly useful when revisiting the solution to understand or explain it better. Each step builds upon the other, bridging understanding and improving your problem-solving skills along the way.