Problem 1 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$\sqrt{169}$$

Step-by-Step Solution

Answer

The value of \(\sqrt{169}\) is 13.

Step 1: Understanding the Problem

We are asked to evaluate the expression \(\sqrt{169}\). This means we need to find the number that, when multiplied by itself, gives 169.

Step 2: Identifying Perfect Squares

Recall the list of perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, and so on. Identify 169 in this list, which corresponds to \(13^2\).

Step 3: Taking the Square Root

The square root operation is essentially the inverse of squaring. Since \(169 = 13^2\), taking the square root gives \(\sqrt{169} = 13\).

Step 4: Final Result

The evaluated expression is \(\sqrt{169} = 13\).

Key Concepts

Perfect Squares

A perfect square is a number that is the square of an integer. In other words, if you multiply an integer by itself, the result is a perfect square. For example, 1, 4, 9, 16, and 25 are all perfect squares. Each of these results from multiplying an integer by itself, like 1 (which comes from 1×1), 4 (from 2×2), and so on.

Understanding perfect squares can make it easier to find square roots and work with quadratic equations. When you know the list of perfect squares, you can quickly recognize them and identify the square root result without needing a calculator.

In the original exercise, identifying 169 as a perfect square helps us immediately know that it is the result of squaring 13, i.e., 13×13 = 169.

Inverse Operations

Inverse operations are pairs of mathematical processes that undo each other. Common pairs of inverse operations include addition and subtraction (e.g., if you add 5 and then subtract 5, you're back to your original number), as well as multiplication and division.

Similarly, squaring a number and taking the square root are inverse operations. Squaring a number means multiplying it by itself, while taking the square root means finding the number that was multiplied. For instance, if we square 13 to get 169, the inverse operation – taking the square root of 169 – brings us back to 13.

This concept is crucial in mathematics because it allows us to solve equations and manipulate expressions. By applying inverse operations, you can isolate variables or simplify complex mathematical problems.

Evaluation of Expressions

Evaluating expressions refers to finding the value of an expression by performing the operations specified in it, often without the aid of a calculator for clear understanding and practice. In the context of our original exercise, evaluating \(\sqrt{169}\) means determining what number, when squared, results in 169.

The process starts with recognizing that 169 is a perfect square – specifically, it's \(13^2\). By understanding this, we can straightforwardly evaluate \(\sqrt{169}\) to be 13.


When evaluating expressions, it’s important to:

• Understand the operations involved, such as addition, subtraction, multiplication, division, or finding roots.
• Know the order of operations – remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication, Division, Addition, Subtraction).
• Convert complex expressions into simpler forms when possible, using mathematical identities and properties.

By keeping these points in mind, students can become proficient in evaluating various kinds of mathematical expressions accurately.