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Chapter 10: Problem 59

find the indicated root, or state that the expression is not a real number. $$\sqrt[4]{1}$$

Short Answer

Expert verified
The 4th root of 1 is 1.

Step by step solution

01

Understanding the Problem

The symbol \(\sqrt[4]{1}\) represents the 4th root of 1. It is the number which when multiplied by itself four times results in 1.
02

Applying the Fourth root operation

The positive 4th root of 1 is 1. This is because \(1^{4} = 1\). The negative root also satisfies because \((-1)^{4} = 1\) as multiplying two negative numbers results in a positive number. However, using the principal root, we use the positive root which is 1.

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Key Concepts

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

Principal Root
When we talk about roots in mathematics, especially fourth roots, the principal root is the one we often focus on. The principal root is essentially the primary or most significant root. For a positive real number, the principal root is also positive.
For example, the principal 4th root of 1, denoted by \( \sqrt[4]{1} \), is 1. This is because \( 1^4 = 1 \). Even though \((-1)^4 = 1\) as well, the principal root conventionally refers to the non-negative option. This helps maintain consistency in solutions across different problems.
Remember:
  • Principal root of even-degree roots (like square roots, fourth roots) is non-negative.
  • It simplifies communication and calculation.
  • For \( \sqrt[n]{x} \), if \(n\) is even and \(x\) is positive, the principal root is positive.
Real Numbers
Real numbers include every number you can imagine on the number line. Real numbers encompass positive numbers, negative numbers, and zero. They can also be fractions or decimals. Essentially, if you can pinpoint it on an infinite number line, it鈥檚 a real number.
When dealing with roots, understanding real numbers becomes crucial. The result of operations like taking the fourth root must be a real number unless stated otherwise. In the original exercise, \( \sqrt[4]{1} \) results in a real number since 1 can be represented on the number line.
Key points about real numbers:
  • They include all rational and irrational numbers.
  • The result of even-dimension roots (like \( \sqrt[4]{1} \)) is real if the number inside the root is positive or zero.
  • Math problems may occasionally ask for solutions that are not real, but these are outside the scope of basic fourth roots like \( \sqrt[4]{1} \).
Negative Root
Negative roots can sometimes be elusive. However, understanding them is important. For a fourth root like \( \sqrt[4]{1} \), there is a negative root in addition to the principal root. This negative root in this context is \(-1\), since \((-1)^4 = 1\). Negative roots are valid because multiplying two negative numbers results in a positive number.
However, it鈥檚 crucial to understand when negative roots are applicable:
  • Odd roots allow negative numbers because \((-x)^n = -1\) and that provides meaningful real results.
  • Even roots on negative numbers lead to non-real numbers, typically involving imaginary numbers, not applicable here with fourth roots of positive numbers.
  • In many contexts, particularly when only real solutions are needed, the principal (positive) root is preferred.
Understanding negative roots helps when specific problems or equations require full solution sets, but often in introductory exercises, focusing on the principal root is sufficient.

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Most popular questions from this chapter

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$$

In Exercises \(137-140\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Some irrational numbers are not complex numbers.

Exercises \(147-149\) will help you prepare for the material covered in the next section. Solve: \(26-11 x=16-8 x+x^{2}\)

In solving \(\sqrt{2 x-1}+2=x,\) why is it a good idea to isolate the radical term? What if we don't do this and simply square each side? Describe what happens.

Solve each equation. $$(x-4)^{\frac{2}{3}}=25$$
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