Problem 19 - Fundamentals of Mathematics

Question

Determine the value of each power and root. \(\sqrt[3]{1}\)

Step-by-Step Solution

Answer

The value of \(\sqrt[3]{1}\) is 1.

Step 1: Identify the Expression

The expression we need to evaluate is the cube root of 1, which is written as \(\sqrt[3]{1}\). A cube root asks which number, when multiplied by itself twice, gives the original number.

Step 2: Understand the Cube Root

The cube root, \(\sqrt[3]{1}\), is the number that when multiplied by itself two more times (meaning three in total), results in 1. We need to find a number \(x\) such that \(x \times x \times x = 1\).

Step 3: Solve for the Cube Root

Given the equation \(x^3 = 1\), we need to determine the value of \(x\) that satisfies this equation. Since \(1^3 = 1\), the value of \(\sqrt[3]{1}\) is 1.

Key Concepts

Understanding How to Evaluate an Expression

When faced with an expression, particularly dealing with roots and powers, the goal is to simplify it to find its value. For example, when you look at the cube root expression \( \sqrt[3]{1} \), it essentially asks you to find the number, which when multiplied by itself three times, returns to the original number 1. This requires an evaluation tactic to simplify the problem and confirm the answer visually and mathematically.

To start:

• Identify the type of expression: Here, it's a cube root.
• Understand that \( \sqrt[3]{1} \) entails finding such a number that fulfilled the equality \( x \times x \times x = 1 \).
• Recognize that the simplest number satisfying this is 1, providing \( 1 \times 1 \times 1 = 1 \).

Once you've simplified the expression into a direct number, your task of evaluating the cube root is complete.

Steps to Solve the Equation

Solving an equation involving roots can initially seem tricky, but it's a logical process. When solving \( x^3 = 1 \), you're essentially trying to find the value of \( x \) that satisfies the equation. This will involve recognizing patterns or utilizing basic arithmetic understanding.

Here's a guide:

• Set up the equation: The equation in the case of \( \sqrt[3]{1} \) is \( x^3 = 1 \).
• Try known solutions: Consider numbers you know might work, like 1, 0, -1, etc.
• Evaluate: For this, you see \( 1 \times 1 \times 1 = 1 \). This satisfies \( x^3 = 1 \). Thus, \( x = 1 \).

In solving such equations, testing known numbers can quickly lead you to the solution, particularly when powers of 1 are involved.

Comprehending Power and Root Operations

Understanding power and root operations is fundamental in algebra. Every number can be expressed with roots, especially cube roots, which simplify repetition of multiplication operations.

Here's what you need to know about operations with cubes:

• A cube, like \( x^3 \), refers to multiplying the number by itself two more times.
• The cube root, like \( \sqrt[3]{x} \), is the inverse operation, taking the number "back" to its root form, which when cubed (multiplied by itself twice), returns the original number.
• Cubes of 1 are unique because \( 1^3 = 1 \), making \( \sqrt[3]{1} \) straightforward to evaluate.

This interplay of inverses, where roots undo powers, is pivotal in simplifying and solving expressions.