Problem 11 - Algebra 2 and Trigonometry

Question

In \(3-14,\) determine whether each of the numbers is rational or irrational. $$ \frac{\sqrt{16}}{2} $$

Step-by-Step Solution

Answer

The number is rational.

Step 1: Simplify the Radical

First, simplify \( \sqrt{16} \). We know that the square root of 16 is 4, because 4 multiplied by 4 equals 16. Thus, \( \sqrt{16} = 4 \).

Step 2: Substitute and Simplify the Expression

Substitute the simplified radical value back into the expression: \( \frac{4}{2} \). Next, divide 4 by 2 to simplify the fraction: \( \frac{4}{2} = 2 \).

Step 3: Classify as Rational or Irrational

The number 2 is a whole number, and it can be expressed as the fraction \( \frac{2}{1} \), where both the numerator and the denominator are integers. Therefore, 2 is a rational number.

Key Concepts

Simplifying Radicals

When dealing with radicals, one of the key steps is simplification. A radical expression involves a root, such as a square root, and simplifying it can help in making calculations easier and clearer. Let's consider \( \sqrt{16} \). The task is to find a value that, when multiplied by itself (squared), gives 16 back. In this case, that value is 4, because \(4 \times 4 = 16\).
Here are a few points to remember when simplifying radicals:

• Find the largest perfect square factor in the radicand (the number inside the square root).
• Rewrite the radical as a product of square roots, separating the perfect square from the other factors.
• Simplify by taking the square root of the perfect square, when possible.

Understanding and practicing these steps can make more complex radical expressions less intimidating and more manageable.

Fraction Simplification

Fractions can often be made simpler by reducing them. Simplifying a fraction involves dividing both the numerator and the denominator by their greatest common divisor (GCD). As seen in the original problem, after simplifying \( \sqrt{16} \) to 4, we were left with the fraction \( \frac{4}{2} \).
Here's how you can simplify fractions effectively:

• Identify the GCD of both the numerator and denominator. In our example, the GCD of 4 and 2 is 2.
• Divide both the numerator and the denominator by the GCD.
• This results in a fraction in its simplest form, \( \frac{2}{1} \), which is equal to 2.

Remember, simplifying fractions not only clarifies the problem but also reveals its essence, making it easier to see if a number is rational or otherwise.

Whole Numbers

Whole numbers are a basic part of mathematics that include all non-negative numbers without fractions or decimals. They range from 0 upwards: 0, 1, 2, 3, and so on. In the given exercise, we simplified \( \frac{4}{2} \) down to 2. Since 2 is a whole number, we can also express it as a fraction \( \frac{2}{1} \), where both the numerator and the denominator are integers.
Key aspects of whole numbers include:

• Always non-negative, meaning they start from zero and increase positively.
• Can be expressed as a fraction \( \frac{n}{1} \), showing that every whole number is, by definition, a rational number.

Whole numbers are fundamental in various mathematical concepts, including counting, ordering, and basic arithmetic operations, and understanding them is essential for recognizing how numbers fit into the broader category of rational numbers.