Problem 4 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. $$\sqrt[4]{16}$$

Step-by-Step Solution

Answer

The fourth root of 16 is 2.

Step 1: Understanding the expression

The expression \( \sqrt[4]{16} \) asks us to find the number that, when raised to the fourth power, equals 16.

Step 2: Considering possible values

We need to consider numbers which when raised to the power of 4 result in 16. Let's start with a simple candidate, 2.

Step 3: Testing our candidate

We test if 2 is the fourth root of 16 by calculating \( 2^4 \). This means multiplying 2 by itself four times: \( 2 \times 2 \times 2 \times 2 = 16 \).

Step 4: Confirming the result

Since \( 2^4 = 16 \), 2 is confirmed to be the fourth root of 16.

Key Concepts

Exponents

Exponents, often referred to as power, describe how many times a number, called the base, is multiplied by itself. For example, in the expression \( 2^4 \), 2 is the base and 4 is the exponent. This means multiplying 2 by itself four times: \( 2 \times 2 \times 2 \times 2 \).
Each time you multiply the base by itself, it is called "raising to a power." This concept is crucial in solving problems involving expressions like \( \sqrt[4]{16} \), where you reverse the process to find out which number, when used as a base with a certain exponent, results in 16.
Key points about exponents include:

• They express repeated multiplication.
• An exponent of 2 is often referred to as "squared," and an exponent of 3 as "cubed."
• Higher exponents involve further multiplications.

Understanding exponents is a stepping stone to mastering mathematical operations involving powers.

Radicals

Radicals are symbols used to denote the root of a number. The most common radical sign is the square root \( \sqrt{} \), but it can have indices to represent other roots, like cube roots \( \sqrt[3]{} \) or fourth roots \( \sqrt[4]{} \). For \( \sqrt[4]{16} \), we are searching for a number that raised to the fourth power equals 16.
Essential components of radicals include:

• The radical symbol (\( \sqrt{} \)) which indicates the root you are finding.
• The index, which is the small number above the radical sign and tells you which root to find (e.g., 4 in \( \sqrt[4]{} \)).
• The radicand, the number inside the radical (e.g., 16 in \( \sqrt[4]{16} \)).

To solve radicals, especially fourth roots like \( \sqrt[4]{16} \), involves reversing exponentiation by testing values.

Problem Solving

Problem solving in mathematics involves identifying and applying appropriate strategies to find solutions. When dealing with a fourth root problem like \( \sqrt[4]{16} \), breaking it into manageable steps can simplify the process.
Here’s a streamlined approach to solving such problems:

• Understand the expression: Determine what the problem asks by analyzing the operation, such as finding a fourth root.
• Consider possible values: Think logically about which numbers, when multiplied to themselves multiple times, could result in the given number.
• Test your candidate: Choose a number to test (e.g., 2) and verify by calculations (like checking if \( 2^4 = 16 \)).
• Confirm and conclude: Once you determine the correct number, re-check your calculations to ensure the result is consistent with the problem.

This structured approach leads to effective problem-solving, ensuring clarity and accuracy in mathematical solutions.