Problem 17 - Fundamentals of Mathematics

Question

Determine the value of each power and root. \(\sqrt[4]{16}\)

Step-by-Step Solution

Answer

The fourth root of 16 is 2.

Step 1: Identify the Problem Statement

We need to find the fourth root of 16, which is represented as \( \sqrt[4]{16} \). This means we are looking for a number which, when raised to the power of 4, equals 16.

Step 2: Recall Fourth Root Definition

The fourth root of a number \( x \), written as \( \sqrt[4]{x} \), is a number \( a \) such that \( a^4 = x \).

Step 3: Make an Initial Guess

Consider some small integers to see if they satisfy the condition. Start with 2, because it’s often a simple guess for powers and roots.

Step 4: Perform Calculation Check for \(a = 2\)

Calculate \( 2^4 \). We have that \( 2 \times 2 = 4 \), \( 4 \times 2 = 8 \), and \( 8 \times 2 = 16 \), so \( 2^4 = 16 \). Thus, 2 is indeed the fourth root of 16.

Step 5: Validate the Result

Verify that no other small integer (besides 2) raised to the fourth power equals 16. Checking integers such as 1 gives \( 1^4 = 1 \) and 3 gives \( 3^4 = 81 \), confirming that 2 is the only valid fourth root of 16 among small integers.

Key Concepts

Fourth Root

When we talk about the 'fourth root' of a number, we're identifying a mathematical operation that reverses the process of raising that number to the fourth power. Just like the square root reverses the multiplication of a number by itself, the fourth root undoes the effect of multiplying a number by itself four times. In notation, the fourth root of a number \( x \) is represented by \( \sqrt[4]{x} \). This means we're looking for a number \( a \) that satisfies:

• \( a^4 = x \)

Let's use 16 as an example. We need to find a number \( a \) such that when we compute \( a \times a \times a \times a \), the result would be 16. Understanding this concept is crucial for delving deeper into roots and powers, and it makes solving these problems methodical and clear.

Integer Powers

Integer powers involve multiplying a number by itself a certain number of times. When we discuss powers in mathematics, we often use terms like 'squared' for the second power, 'cubed' for the third power, and so on. For the fourth power, we simply multiply the number by itself four times. If we have a number \( b \) and want to find its fourth power, we perform the operation \( b \times b \times b \times b \) or \( b^4 \).

• This is what we did in our calculation when checking if 2 was the fourth root of 16.
• We found that \( 2^4 = 2 \times 2 \times 2 \times 2 = 16 \).

Remember that powers are reversible through roots, which makes them an essential concept in understanding the relationship between numbers.

Problem Statement

Identifying the problem statement is the first step in tackling any mathematics problem. A well-defined problem statement lets you know exactly what you're solving for. In this case, the problem statement is to determine the fourth root of 16, written mathematically as \( \sqrt[4]{16} \).

• We are seeking a number, \( a \), that when multiplied by itself four times results in the original number, 16.
• This clear problem definition guides us through the process of verifying possible solutions by raising small integers to the fourth power to see which one equals the original number.

A precise problem statement keeps your calculations focused and ensures you understand the mathematical process required to find the solution.

Calculation Verification

Calculation verification is a critical part of problem solving in mathematics. Once you've proposed a solution, you need to verify it to ensure accuracy. In the context of finding the fourth root of 16, once we guessed that 2 might be our answer, we calculated \( 2^4 \) to confirm that it equals 16.

• Multiply: \( 2 \times 2 = 4 \), \( 4 \times 2 = 8 \), and finally, \( 8 \times 2 = 16 \) confirms \( 2^4 = 16 \).
• Additionally, checking other small integers helps rule out any other candidates, reinforcing that our calculated root is correct and complete.

This kind of thorough checking avoids errors and strengthens your mathematical reasoning by ensuring you return to your work methodically to see that no steps were skipped.