Problem 88 - Introductory Algebra for College Students

Question

Solve each equation .Use a calculator to help with the arithmetic. Check your solution using the calculator. Evaluate: \((-10)^{2} .\)

Step-by-Step Solution

Answer

The result of the expression \((-10)^{2}\) is 100.

Step 1: Understanding the Problem

Our task is to evaluate the expression, \((-10)^{2}\). To do so, we need to follow the order of operations, which indicates that the exponent operation should be done before any other operation.

Step 2: Evaluate the Expression

The operation \((-10)^{2}\) means -10 is to be squared, or alternatively, -10 multiplied by -10. As per the rules of multiplication, a negative times a negative becomes a positive. Hence, \(-10 \times -10 = 100\).

Step 3: Check the Result

A check can be done by comparing the result with the mathematical fact that the square of a number is always positive. Thus, our result 100 is indeed positive, confirming the correctness of our computation.

Key Concepts

Order of Operations

To solve mathematical problems accurately, it's vital to understand the 'order of operations', also known by the acronym PEMDAS. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This rule guides us on which calculation to perform first in a given mathematical expression.

In the given exercise, where we need to evaluate \( (-10)^{2} \), the order of operations dictates that we handle the exponentiation before considering any other arithmetic operations. This is a common area of confusion, as overlooking the order in which operations are performed can lead to incorrect results. A helpful tip is to solve problems step-by-step, writing out each stage of the calculation to avoid errors.

Solving Equations

When solving equations, it's not just about finding an answer; it's about understanding the relationships between variables and constants. Equations can vary in complexity, from simple one-step equations to multi-step ones with various operations. The key is to isolate the variable by performing inverse operations.

However, in our example of \( (-10)^{2} \), we're dealing with a simple equation with no variables to solve for. It's a straightforward evaluation of an expression involving exponentiation. Students often confuse equations with expressions, so it's important to recognize that an equation typically includes an equals sign and a variable, whereas an expression is a series of numbers and operations that represent a value.

Arithmetic Operations

The four basic arithmetic operations are addition, subtraction, multiplication, and division. Each operation has its own set of rules that need to be followed, especially when combining different operations in a single problem.

The example \( (-10)^{2} \) exclusively involves multiplication, as exponentiation is essentially repeated multiplication. It's critical to remember that the result of multiplying two negative numbers is a positive result. Therefore, when we multiply -10 by itself, we obtain a positive number. This example gives a perfect illustration of why understanding arithmetic operations is crucial for solving more complex mathematical problems.