Problem 32 - Algebra 1: Concepts and Skills

Question

Evaluate the variable expression when a = 3 and c = 5. $$ a^{2}+c^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to 34

Step 1: Substitute the given values

First, you need to replace the variables in the expression with the given values. So, replace 'a' with 3 and 'c' with 5 in the expression \(a^{2}+c^{2}\)

Step 2: Evaluation

After substitution, the expression will become \(3^{2}+5^{2}\). Now, you simply have to evaluate this expression. Remember that exponentiation precedes addition.

Step 3: Simplify the expression

Now, simplify the equation. You can do that by performing the exponentiation first: \(3^{2}\) equals 9 and \(5^{2}\) equals 25. Thus, the expression becomes 9 + 25.

Step 4: Final Calculation

Then, add these two values: \(9 + 25 = 34\).

Key Concepts

Substitution in Algebra

Substitution is a crucial concept in algebra. It allows us to replace variables within an expression with actual values. This makes the expression solvable. Consider the expression: \( a^2 + c^2 \). Here, \( a \) and \( c \) are variables. In our example, we know that \( a = 3 \) and \( c = 5 \). By substituting these values directly into the expression, it becomes \( 3^2 + 5^2 \). This approach simplifies the problem and sets the stage for further calculation.

• Always ensure the correct values are substituted for each variable.
• Double-check that each replacement is properly calculated in subsequent steps.

Once substitution is complete, the expression is ready for the next steps in solving.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number to a power. In the expression \( 3^2 + 5^2 \), we have two terms involving exponentiation: \( 3^2 \) and \( 5^2 \). When you "square" a number, you multiply it by itself. So, \( 3^2 \) is simply \( 3 \times 3 \), which equals 9. Similarly, \( 5^2 \) is \( 5 \times 5 \), which equals 25.

• Remember that exponentiation has a higher precedence over other operations like addition or subtraction. Always handle it first in expressions.
• Understand how to simplify different power levels, such as cubing or taking higher powers.

With exponentiation, your expression begins to take shape, showcasing clarity as you move towards final calculations.

Order of Operations

In any mathematical expression, the order of operations is vital. It dictates the sequence in which different parts of the expression should be evaluated to ensure a correct result. For our problem, after substitution, we have \( 3^2 + 5^2 \). Using the rules of the order of operations, known as PEMDAS/BODMAS:

• P/B: Parentheses/Brackets
• E/O: Exponents/Orders
• M-D/B-D: Multiplication and Division
• A-S: Addition and Subtraction

First, solve the exponentiation, \( 3^2 = 9 \) and \( 5^2 = 25 \). Then, proceed to addition: \( 9 + 25 = 34 \).Adhering to this sequence avoids miscalculations, ensuring each step's result is accurate. Knowing these steps helps you systematically approach any algebraic problem you encounter.