Problem 96 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The answer is 13.

Step 1: Solving the Exponent

First, we need to calculate the power in the equation. Compute the squares of 2 and 3: \( 2^2 = 4 \)

Step 2: Solving the Exponent

Now, solve the square of the next number: \( 3^2 = 9 \)

Step 3: Adding the Results

Add the results from Step 1 and Step 2. This will give you the final answer: \( 4 + 9 = 13 \)

Key Concepts

Exponents

Exponents are a mathematical way to express repeated multiplication. When you see an expression like \(2^2\), it means that you need to multiply 2 by itself. In this case, \(2^2 = 2 \times 2 = 4\). Similarly, for \(3^2\), it means \(3 \times 3 = 9\). Exponents are a useful tool in mathematics because they provide a shorthand for expressing and calculating large numbers that would otherwise be cumbersome to write out fully.
Exponents follow certain rules which are important to understand:

• Multiplication of Like Bases: When multiplying numbers with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• Power of a Power: When raising a power to another power, you multiply the exponents: \((a^m)^n = a^{m \times n}\).
• Zero Exponent: Any non-zero number raised to the power of zero is 1: \(a^0 = 1\).

Working with exponents can simplify the process of solving equations, especially when dealing with very large numbers.

Step by Step Solution

Solving equations with a step by step solution is crucial for clarity and accuracy. Here is a breakdown based on the exercise given:

• Step 1: Solve Each Exponent. Start by individually solving each exponent in the equation. Calculate each power, such as \(2^2\) and \(3^2\), separately to understand each component clearly.


• Step 2: Compute and Simplify. After solving individual exponents, you will have simplified values like 4 and 9. This makes the final calculation easier to handle.


• Step 3: Combine Results. Once you've solved each part, add the results together, in this case adding 4 and 9 to get 13. This step finalizes your solution and gives you a clear answer.

This approach ensures that each part of the problem is tackled in a logical progression, minimizing errors and increasing understanding. Following such a structured method, especially with more complex problems, can be immensely helpful.

Addition

Addition is one of the fundamental operations in mathematics, and it is crucial when solving equations like this one. The operation simply involves combining numbers to find their total sum. In the example of adding 4 and 9, you are essentially calculating the total number of units you have by combining two amounts.
Addition has a few properties that are useful in various mathematical contexts:

• Commutative Property: This means that the order in which you add numbers does not affect the sum: \(a + b = b + a\).
• Associative Property: When adding three or more numbers, the sum is the same regardless of how the numbers are grouped: \((a + b) + c = a + (b + c)\).
• Identity Property: The sum of any number and zero is that number: \(a + 0 = a\).

Understanding these properties of addition helps simplify calculations and solve equations more efficiently. It provides a foundation for solving more complex problems by breaking them down into simpler steps.