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Chapter 8: Problem 57

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

Short Answer

Expert verified
The fourth root of 1 is 1.

Step by step solution

01

Identify the Root and the Radicand

Here, the root is 4 (as given by the superscript before the root symbol) and the radicand (the number from which we are finding the root) is 1.
02

Calculate the Fourth Root of 1

Any number (non-zero) to the power of zero equals 1. Therefore, the fourth root of 1 is 1, as \(1^4 = 1\).

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Key Concepts

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

Radicand
The term "radicand" refers to the number or expression located under a radical sign. In our exercise, the radicand is the number 1 found under the fourth root symbol \( \sqrt[4]{1} \).
Understanding the radicand is crucial because it determines what value we're taking the root of.
For example, in the square root symbol \( \sqrt{x} \), \( x \) is the radicand.
  • The radicand can be a positive or negative number, though negative radicands can result in complex numbers.
  • Radicands in real number roots, such as square roots or fourth roots, need special attention to decide if they result in real numbers.
  • In our specific case, the radicand 1 is positive and straightforward, ensuring a real number result.
Fourth Root
The fourth root of a number asks a simple question: "What number multiplied by itself four times equals the radicand?"
This operation is the reverse of raising a number to the fourth power.
In symbolic notation, it's written as \( \sqrt[4]{x} \), where \( x \) is the radicand.
  • Taking the fourth root reverses the process of raising a number to the power of 4. So if \( y^4 = x \), then \( y = \sqrt[4]{x} \).
  • Commonly, in real numbers, the fourth root of 1 is simply 1, since multiplying 1 by itself four times gives 1.
  • The fourth root works similarly to other even roots like the square root, in that it requires the radicand to be non-negative for real number results.
The significant point here is that any number, when multiplied by itself an even number of times, can produce a non-negative result, which is why fourth roots of positive numbers are always real.
Real Numbers
Real numbers are all the numbers that can be found on the number line.
They include both rational numbers, such as integers and fractions, and irrational numbers, which cannot be expressed as a simple fraction.
This broad category is important when discussing roots, as the result of a root (such as the fourth root) is only a real number if certain conditions are met.
  • Real numbers include any number you can think of, except complex numbers (involving the square root of a negative number).
  • For the root to be a real number, the radicand must be non-negative when the root is even.
  • In our exercise, since 1 is a positive real number, calculating its fourth root results in another real number.
Real numbers are foundational in understanding roots because they dictate when an expression will have a real root or when it will stray into complex territories.

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Most popular questions from this chapter

Simplify by first writing the expression in radical form. If applicable, use a calculator to verify your answer. $$81^{-\frac{5}{4}}$$

In Exercises \(75-82\), rationalize each denominator. Simplify, if possible $$\frac{\sqrt{2}}{\sqrt{7}}+\frac{\sqrt{7}}{\sqrt{2}}$$

Simplify each expression. Write answers in exponential form with positive exponents only. Assume that all variables represent positive real numbers. $$\left(\frac{x^{\frac{2}{5}}}{x^{\frac{6}{5}} \cdot x^{\frac{3}{5}}}\right)^{5}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}}=4^{\frac{1}{2}}$$

Without using a calculator, simplify the expressions completely. $$\frac{3^{-1} \cdot 3^{\frac{1}{2}}}{3^{-\frac{3}{2}}}$$
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