Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-10,\) tell whether each represents a number that is rational, irrational, or neither. $$ \sqrt[4]{16} $$

Step-by-Step Solution

Answer

The expression \( \sqrt[4]{16} \) represents a rational number.

Step 1: Understanding the Expression

The expression given is \( \sqrt[4]{16} \). This denotes the fourth root of 16, meaning we need to find a number that when multiplied by itself four times gives 16.

Step 2: Calculate the Fourth Root of 16

To determine \( \sqrt[4]{16} \), we need to identify a number \( x \) such that \( x^4 = 16 \). Multiplying 2 by itself four times, we get:\[ 2 \times 2 \times 2 \times 2 = 16 \]Thus, \( \sqrt[4]{16} = 2 \).

Step 3: Classify the Result

The result \( 2 \) is a whole number. Whole numbers are rational because they can be expressed as a fraction of integers (\( \frac{2}{1} \)). Therefore, \( \sqrt[4]{16} = 2 \) is a rational number.

Key Concepts

Fourth Roots

When we talk about fourth roots, we're discussing the number that, when multiplied by itself four times, produces a given value. For instance, the fourth root of 16 is the number that we multiply by itself four times to get 16. In this exercise, we see that the fourth root of 16 is 2. Let's break it down a bit:

• A fourth root is represented by the symbol \( \sqrt[4]{} \).
• Calculating it involves finding a number \( x \) so that \( x^4 \) equals the number under the root symbol (here, 16).

To figure out \( \sqrt[4]{16} \), we check if 2 works by evaluating \( 2 \times 2 \times 2 \times 2 = 16 \). Therefore, \( \sqrt[4]{16} = 2 \). This is one example, and knowing how to find roots is a key math skill. Try practicing with different numbers to get the hang of it!

Whole Numbers

Whole numbers include a straightforward range of numbers, starting from zero and going upwards to positive infinity. They are numbers without any fractions or decimals involved. Here are the main things to remember:

• Whole numbers begin with 0, so 0, 1, 2, 3, and so on are all whole numbers.
• The number 2, which we found as the fourth root of 16 in our example, is a whole number.
• Whole numbers do not include negative numbers or fractions.

The simplicity of whole numbers makes them easy to understand and work with. This is why identifying whether a number is a whole number can help classify it in different math problems!

Classification of Numbers

Classifying numbers helps us understand their properties and how they interact with other numbers. In mathematics, classifications range from whole numbers to more complex ones like rational and irrational numbers.

• Rational numbers are numbers that can be expressed as the quotient of two integers (like 2 which is \( \frac{2}{1} \)).
• Irrational numbers are numbers that cannot be written as simple fractions. For example, \( \sqrt{2} \) is irrational because it cannot be exactly expressed as a fraction.
• Whole numbers are part of rational numbers because you can express them as a fraction with a denominator of 1.

In our exercise, the fourth root of 16 turned out to be 2, which is a rational number since it fits the criteria of a fraction \( \frac{2}{1} \). Understanding these categories helps you quickly figure out the nature of numbers you're working with in math problems.