Problem 80 - Algebra 2

Question

Find each indicated root if it is a real number. $$ \sqrt[4]{16} $$

Step-by-Step Solution

Answer

The fourth root of 16 is 2

Step 1: Understand the problem

We are asked to find the fourth root of 16, which means we have to find a number that when raised to the power 4, gives us 16. This can be noted symbolically as \(x^4 = 16\).

Step 2: Finding the root

When we find the roots, we have two possible solutions, one is a positive root and the other is a negative root. With fourth roots, we just consider the positive roots. We start with the smallest positive integer 1 and keep increasing it. We know that \(2^4 = 16\) so the fourth root of 16 is 2.

Key Concepts

Real Numbers

In mathematics, real numbers cover a broad spectrum of numbers we use in everyday life, including whole numbers, fractions, decimals, and more. Real numbers are critical in problem-solving because they

• encompass all rational and irrational numbers
• allow us to express exact values
• provide a foundation for performing arithmetic operations

In the context of finding roots, we often deal with real numbers because they give us tangible values. For instance, when working with fourth roots like in the exercise, the solution is a specific, easily understandable real number. Here, the fourth root of 16 being 2 is a clear example of a real number solution.

Root Finding

The concept of root finding is crucial in algebra and calculus because it involves determining values that satisfy certain equations. A root of a number is a value which, when raised to a certain power, returns the original number. The process of root finding can be simplified by:

• Identifying the power given (e.g., square root, fourth root)
• Understanding the relationship between powers and roots
• Using logical guess-and-check methods or algebraic techniques

Root finding is not just about calculations but also about reasoning through possible solutions. In our example, finding the fourth root of 16 required identifying that switching the order of 2 and 4 gave us the target number.

Exponents

Exponents indicate how many times a number, known as a base, is multiplied by itself. They are fundamental in expressing repeated multiplication compactly and are central to understanding roots. Here's what makes exponents easy:

• The exponent tells you exactly how many times to multiply the base
• They are pivotal in determining roots like squares and fourth roots

For the problem at hand, moving from understanding that the fourth power of 2 is 16 ( $2^4 = 16$) helps us deduce the fourth root. Recognizing these reciprocal relationships between powers and roots simplifies finding solutions in algebraic equations.

Positive Roots

Positive roots are the non-negative numbers that satisfy the root condition of a given number and power. While some roots have both positive and negative solutions, we often focus on positive roots for simplicity and clarity. When calculating positive roots:

• Consider roots in their simplest form
• Focus on positive integer solutions unless otherwise stated

In the context of the fourth root of 16, while mathematically both negative and positive roots exist, we focus on the simplest positive integer, which is 2. This emphasizes the importance of recognizing patterns that yield clear, real solutions.