Problem 71 - Elementary and Intermediate Algebra

Question

Simplify each cube root. See Example \(6 .\) $$ \sqrt[3]{1} $$

Step-by-Step Solution

Answer

The cube root of 1 is 1.

Step 1: Understand the Cube Root

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. Mathematically, if \( x^3 = a \), then \( x = \sqrt[3]{a} \).

Step 2: Simplify \( \sqrt[3]{1} \)

Since we know \( 1^3 = 1 \), the cube root of 1 is the number that when multiplied three times gives 1. Thus, \( \sqrt[3]{1} = 1 \).

Step 3: Verify the Result

To confirm, calculate \( 1 \times 1 \times 1 = 1 \). This shows that if the cube root of 1 is 1, multiplying it by itself three times indeed gives 1.

Key Concepts

Understanding and Simplifying Radicals

Radicals might seem daunting at first, but they simply involve finding the root of a number. When we look at simplifying cube roots like \( \sqrt[3]{1} \), we are essentially searching for a number which, when multiplied by itself three times, provides the original number. So, in this example, we seek a value that returns 1 when cubed.
To simplify a radical effectively:

• Identify the base number you're working with, which is 1 in our case.
• Determine if there is a number that can multiply by itself three times to get the base.
• If such a number exists, that's your simplified radical.

Cube roots of perfect cubes are integers. This is why \( \sqrt[3]{1} = 1 \) is a simple expression, as one in its cube form remains unchanged at 1.

Using Mathematical Reasoning

Mathematical reasoning is critical in understanding and simplifying cube roots. It's not just about seeing a number and jotting down an answer.
This reasoning involves:

• Logically thinking through the multiplication process (What times what equals the original number?)
• Analyzing patterns within numbers (Recognizing that 1 always results in 1 when multiplied by itself any number of times).
• Utilizing mathematical principles such as identity (1 times anything is itself).

Reasoning helps us see that since \(1^3 = 1\), it fulfills the condition necessary for \( \sqrt[3]{1} \) to equal 1. Reasoning this way strengthens comprehension and aids conceptual learning.

Verification of Solutions

After simplifying a cube root, verifying your solution ensures accuracy and understanding. This isn't just about intuition or guesswork; it's a fundamental part of mathematical practice.
Verification involves subtly reapplying the cube root conditions:

• Take your simplified result—in this case, 1.
• Compute its cube — \( 1 \times 1 \times 1 \).
• Confirm this product equals the original number (1), proving the cube root's accuracy.

Verification gives confidence in the solution. Proving on paper that \( \sqrt[3]{1} = 1 \) by continually deriving the original number affirms our reasoning and methods, providing solid evidence of understanding.