Problem 9 - Algebra 1

Question

Evaluate the expression when \(x=3\) $$ (x+1)^{3} $$

Step-by-Step Solution

Answer

The value of the expression when \(x = 3\) is 64.

Step 1: Substituting

Substitute the value \(x = 3\) in the given expression. This will give us \((3 + 1)^{3}\).

Step 2: Simplifying Inside the Parentheses

Inside the parentheses, we have a simple addition of 3 and 1 which gives 4. This results in the expression \(4^{3}\).

Step 3: Evaluating exponent

Evaluate \(4^3\). 4 cubed equals 64.

Key Concepts

Substitution Method

The substitution method is a fundamental concept in algebra that involves replacing variables in an expression with their given numeric values. This practice helps simplify the expression, enabling us to evaluate or solve it with ease. In our exercise, the expression given was \((x+1)^3\) with a known value for the variable \(x\). Here's how we used the substitution method:

• First, identify the variable(s) in the expression. In our example, it's \(x\).
• Next, replace \(x\) with the given value, which is 3 in this case.
• Once substitution is complete, rewrite the expression so that all occurrences of \(x\) are replaced by the number 3. This results in the expression \((3+1)^3\).

By practicing substitution, you simplify your calculations significantly and make solving the problem much more straightforward. It's a handy tool for turning algebraic expressions into numerical calculations.

Order of Operations

When evaluating algebraic expressions, the order of operations is crucial. It ensures that expressions are simplified and evaluated correctly. The commonly used acronym PEMDAS stands for: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Let's apply this to our expression:

• First, evaluate any expressions within parentheses. In \((3+1)^3\), calculate \(3+1\), which is 4.
• Next, handle the exponents. Once inside the parentheses is simplified, you're left with \(4^3\).
• Since there are no multiplication, division, addition, or subtraction operations left, you can focus on evaluating \(4^3\).

Following the correct order ensures accuracy, preventing mistakes that can occur when calculations are done in a different sequence. The order of operations simplifies complex expressions step-by-step, one operation at a time.

Exponents

Exponents are a key concept in algebra representing repeated multiplication of a number by itself. The expression \(b^n\) or base \(b\) raised to the power of \(n\) means multiplying \(b\) by itself \(n\) times. In the exercise, the simplified expression was \(4^3\), which is evaluated as follows:

• Write the base number (4) the number of times indicated by the exponent (3). So: \(4 \times 4 \times 4\).
• Perform the multiplication in sequence. First, multiply the first two 4s: \(4 \times 4 = 16\).
• Next, multiply the result by the final 4: \(16 \times 4 = 64\).

Understanding exponents allows you to efficiently compute large numbers and helps in simplifying polynomial expressions. By mastering the basics of exponentiation, solving problems involving powers becomes second nature.