Problem 9 - Fundamentals of Mathematics

Question

Determine the value of each expression. \(\sqrt[3]{27}\)

Step-by-Step Solution

Answer

The value is 3.

Step 1: Understand the Expression

The task is to find the cube root of 27, which is the number that, when multiplied by itself three times, gives 27.

Step 2: Identify Perfect Cube

Recognize that 27 is a perfect cube. Specifically, 27 is equal to 3 raised to the power of 3, or \(3^3 = 27\). This means that 3 is the cube root of 27.

Step 3: Apply the Cube Root Property

Since 27 = \(3^3\), then its cube root can be written as \(\sqrt[3]{3^3}\). The cube root and the cube exponent cancel each other out, leaving us with 3.

Key Concepts

Perfect Cubes

The concept of perfect cubes is essential in understanding cube roots. A perfect cube is a number that can be expressed as the product of an integer multiplied by itself three times. For example, in the number 27, we multiply 3 by itself three times:

• First multiplication: 3 × 3 = 9
• Second multiplication: 9 × 3 = 27

This confirms that 27 is a perfect cube because it equals to \(3^3\). Recognizing perfect cubes is crucial as it simplifies the process of finding their cube roots, as perfect cubes result in whole numbers after cube root extraction.

Exponentiation

Exponentiation is a fundamental mathematical operation involving powers. When a number is raised to an exponent, it means the number is multiplied by itself a certain number of times indicated by the exponent. For example, \(3^3\) means 3 is used as a factor three times:

• The base is 3.
• The exponent is 3.
• The expression \(3^3\) equals 3 × 3 × 3 = 27.

In the context of cube roots, recognizing the base and exponent helps understand why the expression \(\sqrt[3]{3^3}\) simplifies directly to 3. Exponentiation in mathematical contexts results in perfect cubes, simplifying root calculations.

Mathematical Operations

Understanding mathematical operations, especially involving roots and powers, is integral to solving expressions like finding cube roots. The cube root operation is essentially the inverse of exponentiation by 3. In mathematical terms, if a number, say 27, is equal to \(a^3\), then the cube root \(\sqrt[3]{27}\), equals \(a\). The operation manipulates the expression:

• Apply the cube root to \(27 = 3^3\).
• The cube root cancels out the power of 3, resulting in 3.

Understanding these operations streamlines the process by using the cube root to "undo" the cube of a number. This coordination between powers and roots exemplifies simplicity in complex calculations, converting lengthy multiplication processes into shorter, meaningful solutions.