Problem 8 - Algebra and Trigonometry

Question

Evaluate each expression indicate that the root is not a real number. $$ \sqrt{144+25} $$

Step-by-Step Solution

Answer

The square root of \(144 + 25\) is \(13\).

Step 1: Identify the numbers under the square root

Identify that there are two numbers under the square root, which are \(144\) and \(25\).

Step 2: Perform the Addition

Add the numbers together. This can be done easily. \(144 + 25 = 169\).

Step 3: Calculate the Square Root

The square root of \(169\) is \(13\) because \(13 * 13 = 169\). If the number under the square root had been negative, the root would not be a real number.

Key Concepts

Real Numbers

Real numbers are a fundamental concept in mathematics. They include a wide range of numbers that can be either rational or irrational.
Rational numbers are numbers that can be expressed as a fraction, like 1/2 or 6. On the other hand, irrational numbers cannot be expressed as a simple fraction, such as \(\pi\) or \(\sqrt{2}\).
Real numbers are used everywhere in mathematics and everyday life. They cover:

• Whole numbers like 0, 1, 2, 3.
• Fractions like 3/4 or 2.75.
• Decimals, whether terminating or repeating, e.g., 0.75 or 0.333...
• Irrational numbers like \(\sqrt{2}\) or \(\sqrt{3}\).

When working with real numbers in expressions, such as a square root, it is important to ensure the number inside is not negative, as this would not result in a real number. Instead, it would result in what is called an imaginary number. In our case, numbers like \(144\) and \(25\) are real and positive, allowing us to proceed without issues.

Addition

Addition is one of the four basic operations in arithmetic, alongside subtraction, multiplication, and division. It involves combining two or more numbers to find their total value.
When adding numbers:

• Make sure all numbers are accurately lined up, especially with decimal numbers—it helps ensure precision.
• You can break down the addition into simpler steps if it’s easier, for example, adding by tens first.

In our example, we have\[144 + 25 = 169\]This is a straightforward addition. It is crucial that this step is done correctly because any mistake here will impact all further calculations. Upon correctly adding these real numbers under the square root, you get a new number, 169, which is also a real number.

Square Roots of Perfect Squares

A perfect square is a number that can be expressed as the square of an integer. For example, \(169\) is a perfect square because it can be written as\[13 \times 13 = 169\]Square roots "undo" squaring. They determine what number, when multiplied by itself, results in the given number. When dealing with perfect squares, the square root results in an integer, which is straightforward to compute.
In the exercise, we calculated\[\sqrt{169} = 13\]This demonstrates finding the square root of a perfect square. It helps confirm calculations are correct and the number is real and positive. Understanding how to recognize and work with perfect squares is a valuable skill in simplifying mathematical expressions.