Problem 45 - Algebra 1

Question

Evaluate the expression for the given values of the variables. $$ (d-3)^{2} \text { when } d=13 $$

Step-by-Step Solution

Answer

The expression \((d-3)^{2}\) evaluates to 100 when \(d=13\).

Step 1: Substitute the value of the variable

Substitute the given value \(d = 13\) in the expression \((d-3)^{2}\). So the expression becomes \((13-3)^{2}\).

Step 2: Perform the operation inside the parentheses

Subtract 3 from 13 which gives us 10. So, the expression becomes \(10^{2}\).

Step 3: Square the result to get the final answer

The square of 10 is 100. So, \((d-3)^{2}\) simplifies to 100 when \(d=13\).

Key Concepts

Substitution in Algebra

Substitution is a fundamental technique in algebra where we replace variables with their numerical values. In the provided exercise, we're given the expression \( (d-3)^{2} \) and the value of \( d = 13 \). By substituting 13 for \( d \), the expression \( (13-3)^{2} \) is obtained. This step transforms an abstract algebraic expression into a specific numerical example, allowing for clear and straightforward calculation.

Why Substitute?

Substitution simplifies complex algebraic equations by turning them into easier arithmetic problems. It also enables us to evaluate expressions for different variable values without modifying the initial equation's structure.

Common Pitfalls

It's important, however, to substitute precisely and handle the resulting numerical expression carefully, to avoid simple mistakes that could lead to incorrect results.

Order of Operations

The order of operations is a rule that tells us the sequence in which we should solve different parts of a mathematical expression. It is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Applying PEMDAS

When evaluating \( (d-3)^{2} \) after substitution, we first address the operation inside the parentheses \( (13-3) \) to obtain 10. Following the order of operations, we then square the result because exponents come right after parentheses in PEMDAS. This strict adherence ensures consistency and accuracy in solving mathematical expressions.

Common Missteps

Students often make the error of performing operations out of the prescribed order, which can drastically alter the final result. Remembering and applying PEMDAS is crucial for the correct evaluation of expressions.

Squaring Numbers

Squaring a number means multiplying the number by itself. The square of a number is represented by the number raised to the power of two, like \(10^{2}\). In our exercise, once we have simplified the expression to \(10^{2}\), we calculate the square of 10.

Visualizing Squares

To visualize squaring, imagine a square with sides of length 10. The area of this square, representing \(10^{2}\), would be 100 square units—the result of our original expression.

• The square of a positive number is always a positive number.
• The squaring process is a powerful tool in algebra, as it can simplify expressions and highlight the properties of numbers.

Real-World Applications

Squaring numbers is not just a mathematical exercise; it has practical applications in geometry, physics, and finance, where it helps to model relationships and calculate various measurements.