Problem 15 - Algebra 2 and Trigonometry

Question

In \(11-38,\) evaluate each expression in the set of real numbers. $$ \sqrt{169} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We are asked to evaluate the expression \( \sqrt{169} \) which means we need to find the principal square root of 169.

Step 2: Recall the Definition of Square Root

The square root of a number \( x \) is another number \( y \) such that \( y^2 = x \). In our problem, we want \( y^2 = 169 \).

Step 3: Identify Known Perfect Squares

Since \( 169 = 13 \times 13 \), we can recognize that 169 is a perfect square of 13.

Step 4: Evaluate the Square Root

Given that 13 squared equals 169, the square root \( \sqrt{169} = 13 \).

Key Concepts

Real Numbers

Real numbers are all the numbers you have likely encountered in your everyday life. They include both rational numbers, like fractions and integers, and irrational numbers, which cannot be neatly expressed as fractions. Think of real numbers as the entire set of points on an infinitely long number line.

• Real numbers encompass values that can be positive, negative, or zero.
• They also include decimal numbers, whether they terminate or repeat, like 0.8 or 2.3333...
• Real numbers are used to denote temperature, height, time, and much more in practical situations.

This extensive set of numbers provides the basis for understanding square roots. A square root is a real number when it results in decay from a non-negative real number.

Perfect Squares

Perfect squares are numbers that result from squaring an integer. Essentially, if you multiply an integer by itself, you get a perfect square. For example, if you square 4,

• which is calculated as 4 x 4, you get 16. Here, 16 is a perfect square.
• Similarly, 169 is a perfect square because it results from multiplying 13 by itself.
• This close relationship between numbers makes it easier to evaluate square roots when dealing with perfect squares.

Understanding perfect squares is crucial because they provide a direct route to calculating square roots, as illustrated by finding that the square root of 169 is 13.

Principal Square Root

The principal square root of a number is the non-negative root you'd get from a square root operation. It is defined as the square root that is not negative, ensuring consistency in mathematical results.

• When you calculate the square root of 169, expecting to find the principal square root, you aim to find the non-negative value that when squared equals 169.
• In our case, the principal square root of 169 is 13 because 13 x 13 equals 169.
• This avoids any confusion that could arise from the fact that squaring a negative number also results in a positive product. Hence, while both 13 and -13 are square roots of 169, we refer to 13 as the principal square root.

Understanding the concept of a principal square root ensures clarity when interpreting square root results in mathematical exercises and real-world applications.