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Chapter 9: Problem 71

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

Short Answer

Expert verified
The cube root of 1 is 1.

Step by step solution

01

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 \( x = \sqrt[3]{a} \).
02

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, \( \sqrt[3]{1} = 1 \).
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攊n this case, 1.
  • Compute its cube 鈥� \( 1 \times 1 \times 1 \).
  • 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.

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Most popular questions from this chapter

Simplify each expression. All variables represent positive real numbers. See Example 7. $$ \left(\frac{25}{49}\right)^{-3 / 2} $$

Simplify each expression. All variables represent positive real numbers. $$ n^{1 / 5}\left(n^{2 / 5}-n^{-1 / 5}\right) $$

Simplify each radical expression, if possible. Assume all variables are unrestricted. $$ \sqrt[3]{1,000 a^{6} b^{6}} $$

The function \(s(g)=\sqrt[3]{\frac{g}{7.5}}\) determines how long (in feet) an edge of a cube-shaped tank must be if it is to hold \(g\) gallons of water. What dimensions should a cube-shaped aquarium have if it is to hold 1,250 gallons of water?

Use a calculator to evaluate each expression. Round to the nearest hundredth. See Using Your Calculator: Rational Exponents. $$ (-1,000)^{3 / 5} $$
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