Problem 47 - Algebra for College Students

Question

Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. \((x-y)^{2}, \quad x=5\) and \(y=-3\)

Step-by-Step Solution

Answer

The expression evaluates to 64.

Step 1: Substitute the Values into the Expression

Start by substituting the given values of the variables into the expression. The original expression is \((x-y)^{2}\). Substitute \(x = 5\) and \(y = -3\) into the expression to get the following: \((5 - (-3))^{2}\).

Step 2: Simplify Inside the Parentheses

Simplify the expression inside the parentheses. The expression \(5 - (-3)\) simplifies to \(5 + 3\) because subtracting a negative is the same as adding a positive. So, we have \((5 + 3)^{2}\), which simplifies further to \(8^{2}\).

Step 3: Calculate the Square

Calculate the square of the simplified term. Here, \(8^{2}\) means \(8\) multiplied by itself, which is \(8 \times 8 = 64\).

Step 4: Finalize the Answer

Having calculated the square, the evaluation of the expression \((x-y)^{2}\) for \(x=5\) and \(y=-3\) gives us the final result of \(64\).

Key Concepts

Substitution

Substitution is the process of replacing variables in an algebraic expression with their given numerical values. It is a fundamental step when evaluating expressions, ensuring you work with specific numbers rather than variables. To substitute:

• Identify the variables in the expression.
• Replace each variable with its assigned value.
• Re-evaluate the expression using these values.

In this exercise, we replaced the variables in \((x-y)^2\) with \(x = 5\) and \(y = -3\). After substitution, the expression looked like this: \((5 - (-3))^2\). By understanding this process, you naturally simplify further calculations. Remember, careful substitution sets the stage for a correct final solution.

Simplifying Expressions

Simplifying expressions is about combining like terms and making an expression easier to work with. When expressions involve numbers and operations, simplification often includes working through them in a logical order. For this example:

• Start inside the parentheses: calculate \(5 - (-3)\).
• Subtracting a negative number means adding its absolute value. Therefore, \(5 - (-3)\) becomes \(5 + 3\).

After simplifying, the expression becomes \(8^2\). Simplifying ensures that calculations are straightforward and errors are minimized. This is an essential skill for algebra, helping you tackle more complex problems with ease.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. The exponent tells us how many times the base number is used as a factor. For example, \(8^2\) indicates that 8 should be multiplied by itself. Here's how you calculate exponents:

• Recognize the base and the exponent number.
• Multiply the base by itself as many times as indicated by the exponent.
• \(8^2\ = 8 \times 8 = 64\), which is the final calculation in our example.

Understanding exponents allows you to perform calculations in a structured way, especially when expressions involve powers. Mastering this concept is key in algebra and beyond.