Problem 47 - Intermediate Algebra

Question

Evaluate the algebraic expressions for the given values of the variables. $$ (x-y)^{2}, \quad x=5 \text { and } y=-3 $$

Step-by-Step Solution

Answer

The expression evaluates to 64.

Step 1: Substitute Values into the Expression

Take the given expression \((x-y)^{2}\) and substitute \(x = 5\) and \(y = -3\). This gives us \((5 - (-3))^{2}\).

Step 2: Simplify the Inner Expression

Simplify the expression inside the parentheses: \(5 - (-3) = 5 + 3 = 8\). So now we have \(8^{2}\).

Step 3: Apply the Exponentiation

Compute the square of 8: \(8^{2} = 64\).

Key Concepts

Substitution Method

The substitution method is a powerful approach in algebra that involves replacing variables with their corresponding numerical values. This method helps simplify and solve equations or expressions. In our example, the expression is \( (x-y)^{2} \) with given values \( x = 5 \) and \( y = -3 \). By substituting these values, the expression becomes \( (5 - (-3))^{2} \).

• Why Substitute? Using substitution provides clarity by turning an abstract expression into a numerical one.
• Eliminate Variables: It helps to eliminate variables, thus simplifying calculations.

The process involves plugging in the values directly into the expression wherever the variables appear, ensuring to follow the algebraic operations involved.

Simplifying Expressions

Simplifying expressions is an essential step in solving algebraic problems. It involves performing arithmetic operations to make the equation more manageable. In the expression \( (5 - (-3))^{2} \), we begin by focusing on simplifying the part inside the parentheses.
When we have a minus sign followed by a negative number, it transforms into addition:

• Minus a Negative: \( 5 - (-3) \) simplifies to \( 5 + 3 \).
• Compute the Sum: This results in \( 8 \).

After simplification, the expression becomes \( 8^{2} \). This step reduces complications when dealing with more complex expressions, paving the way for easier calculations.

Exponentiation

Exponentiation refers to the mathematical operation involving powers. It's where a number, termed the base, is raised to an exponent or power, indicating how many times the base is multiplied by itself. In the expression \( 8^{2} \), 8 is the base and 2 is the exponent.

• Understanding Exponents: \( 8^{2} \) means multiplying 8 by itself once, yielding \( 8 \times 8 \).
• Calculate the Result: This multiplication equals \( 64 \).

Exponentiation is common in various mathematical contexts, simplifying the representation of repeated multiplication. This procedure is crucial in completing our original problem by calculating the final value of the expression.