Problem 31 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The evaluated expression results in 64.

Step 1: Substitute the Values

Replace a and c in the expression \((a+c)^{2}\) with 3 and 5 respectively. So the expression to compute becomes \((3 + 5)^{2}\).

Step 2: Simplify the Parentheses

Perform the operation inside the parentheses. The parenthesis contains an addition operation 3 + 5. This simplifies to \((8)^{2}\).

Step 3: Square the Result

The last operation is squaring the obtained value i.e., \(8^{2}\). The square of 8 is 64.

Key Concepts

Substitution Method

Understanding the substitution method is essential when working with algebraic expressions involving variables. This method involves replacing variables in an equation with their given numerical values. In our original exercise, we were tasked with evaluating the expression \((a+c)^2\) when \(a=3\) and \(c=5\).

This is how the substitution method plays out:

• Identify the variables in your algebraic expression.
• Use the given values to replace each variable accordingly. In other words, substitute \(a\) with 3 and \(c\) with 5.
• This transforms the expression to \((3+5)^2\).

Substituting the values is a straightforward step, but it's crucial for turning a complex expression into something you can calculate. Remember, accuracy during substitution will ensure the correctness of the entire process.

Order of Operations

In mathematics, the order of operations is a set of rules to determine the sequence in which calculations should be performed. Adhering to these rules is important to avoid incorrect results. The proper order can be remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Let's explore this using our example:

• Step 1: "Parentheses". First, solve the operations inside parentheses. In \((3+5)^2\), start by adding the numbers inside the brackets: \(3 + 5 = 8\).
• Step 2: "Exponents". With \(8^2\) remaining, we then handle the exponent: calculate \(8^2\) which results in 64.

Following this specific order ensures all calculations yield accurate and consistent results. Make sure to pause during each step, double-checking each part of the expression before moving on.

Exponents

Exponents are a fundamental part of algebra and are used to denote repeated multiplication of a number by itself. In the expression \((8)^{2}\), the exponent is 2, telling us to multiply 8 by itself.

Here's a closer look at how to work with exponents:

• The base is the number being multiplied — here, it's 8.
• The exponent is the superscripted number — here, 2 — which indicates how many times the base is multiplied by itself.
• To evaluate \(8^2\), compute \(8 \times 8\), which equals 64.

Understanding how exponents function makes it easier to expand any expression that involves exponents. When tackling these problems, ensure that your calculations are precise to avoid errors in your final answer.