Problem 9 - Fundamentals of Mathematics

Question

Find the following roots using only a knowledge of multiplication. \(\sqrt[3]{64}\)

Step-by-Step Solution

Answer

The cube root of 64 is 4.

Step 1: Understand the Problem

We need to find the cube root of 64. A cube root of a number is a value that, when multiplied by itself three times, equals the original number. So essentially, we are looking for a number that, when used in the expression \( n \times n \times n \), equals 64.

Step 2: Identify Possible Values

Let's consider whole numbers that when cubed, might equal 64. Begin with smaller numbers and work upwards: 1, 2, 3, 4, etc. We will multiply each number by itself twice (i.e., cube it) to see if the result equals 64.

Step 3: Multiply and Check

Start with small whole numbers:- For 1: \( 1 \times 1 \times 1 = 1 \), not 64.- For 2: \( 2 \times 2 \times 2 = 8 \), not 64.- For 3: \( 3 \times 3 \times 3 = 27 \), not 64.- For 4: \( 4 \times 4 \times 4 = 64 \), perfect match!Since \( 4 \times 4 \times 4 \) equals 64, 4 is the cube root of 64.

Step 4: Confirm the Calculation

To ensure this solution is correct, check the multiplication for 4 again:Calculate \( 4 \times 4 = 16 \).Then calculate \( 16 \times 4 = 64 \).The calculation confirms that multiplying 4 by itself three times results in 64.

Key Concepts

Multiplication

Multiplication is one of the basic operations in mathematics, and it involves adding a number to itself a specific number of times. When we talk about the cube of a number, we mean multiplying that number by itself two more times. For example, if we are finding the cube of 2, we calculate it as follows:

• 2 multiplied by 2 equals 4.
• Then, multiply 4 by 2 to get 8.

So, the expression for cubing a number can be written as: \( ext{Number} imes ext{Number} imes ext{Number} \). Most often, this is condensed in expression using the exponent form, such as \( 2^3 = 8 \). This approach requires practicing multiplying numbers in sequence to understand their relationships and result in efficient calculations.

Whole Numbers

Whole numbers are the set of numbers that include all the natural numbers plus zero. These are numbers without fractions or decimals and are used frequently in everyday tasks and calculations. For example, when determining the cube root, as seen in this exercise, you only consider whole numbers:

• 0, 1, 2, 3, 4, 5, etc.

When trying to find the cube root of 64 in whole numbers, we had to test a series of whole numbers starting from 1. By cubing these whole numbers, we determined that 4 cubed equals 64. Using whole numbers simplifies calculations, ensuring that the results are consistent and easy to verify in exercises like these.

Cube Root Calculation

The cube root of a number is a smaller number that, when raised to the power of three, reproduces the original number. Knowing the concept of cube roots helps to reverse the cubing process, where instead of multiplying a number by itself three times, we find which number it could be. Here’s how it was applied:

• Start with 1: \( 1 imes 1 imes 1 = 1 \), clearly not 64.
• Try 2: \( 2 imes 2 imes 2 = 8 \), still not 64.
• Continue with trials until reaching the correct cube root for 64, which was verified with 4: \( 4 imes 4 imes 4 = 64 \).

To verify this, we can re-calculate \( 4 imes 4 = 16 \) and then multiply \( 16 imes 4 = 64 \). This step-by-step approach ensures confident and accurate results in cube root calculations.