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Chapter 8: Problem 81

Simplify using absolute value signs as needed. (a) \(\sqrt[5]{-32}\) (b) \(\sqrt[8]{-1}\)

Short Answer

Expert verified
Part (a) is -2. Part (b) is not a real number.

Step by step solution

01

Understand the Problem

The task involves simplifying roots with negative numbers as radicands, using absolute value signs if required.
02

Evaluate Part (a)

For \(\root[5]{-32}\), identify that \(-32\) can be written as \((-2)^5\). Since the index of the root (5) is odd, the root of a negative number is allowed and will be negative. Therefore, \(\root[5]{(-2)^5} = -2\).
03

Evaluate Part (b)

For \(\root[8]{-1}\), identify that \(-1\) can be written as \((-1)^8\). Since the index of the root (8) is even, the root of a negative number is not allowed and will be imaginary. However, for real roots, an absolute value must be used. Thus, \(\root[8]{-1} = \text{not a real number}\) unless specifically considering complex roots.

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Key Concepts

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

Absolute Value
The absolute value of a number is its distance from zero on the number line, regardless of the direction. In other words, it's always a non-negative number. The absolute value of a number is denoted by vertical bars around the number, like this: \(|x|\). For example, \(|-5| = 5\) and \(|5| = 5\). The concept of absolute value is important when simplifying roots, especially with even indices and complex numbers.
Radicands
A radicand is the number under the root symbol. For example, in \(\sqrt{9}\), the radicand is \({9}\). Radicands can be positive or negative, but this affects how we simplify them, especially when considering complex or real root solutions. When working with roots, always examine the radicand to determine if the results will be real or complex.
Odd and Even Roots
Roots can be classified based on whether their indices (the small number outside the root symbol) are odd or even. This classification determines how we simplify them. For example, \(\sqrt[5]{-32}\) involves an odd root, allowing us to have negative results, like \(-2\). On the other hand, for even roots, such as \(\sqrt[8]{-1}\), the result could be complex unless the radicand is non-negative. Understanding whether a root is odd or even helps in determining the nature of the solution, whether real or complex.
Complex Numbers
Complex numbers arise when dealing with even roots of negative numbers. A complex number has a real part and an imaginary part and is written in the form \(a + bi\), where \(i\) is the imaginary unit \(i^2 = -1\). For instance, \(\sqrt[8]{-1}\) results in a complex number because the index is even, and the radicand is negative. In simpler terms, roots of negative numbers are real if the index is odd and complex if the index is even. Understanding complex numbers is essential for delving into advanced root problems and simplifying them correctly.

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Most popular questions from this chapter

Use the Quotient Property to simplify square roots. (a) \(\sqrt{\frac{28 p^{7}}{q^{2}}}\)(b) \(\sqrt[3]{\frac{81 s^{8}}{t^{3}}}\) (c)\(\sqrt[4]{\frac{64 p^{15}}{q^{12}}}\)

Use the Quotient Property to simplify square roots. (a) \(\frac{\sqrt{50 m^{7}}}{\sqrt{2 m}}\)(b) \(\sqrt[3]{\frac{1250}{2}}\)(c)\(\sqrt[4]{\frac{486 y^{9}}{2 y^{3}}}\)

Write with a rational exponent. (a) \(\sqrt[4]{r^{7}}\)(b) \((\sqrt[5]{2 p q})^{3}\)(c)\(\sqrt[4]{\left(\frac{12 m}{7 n}\right)^{3}}\)

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Use the Quotient Property to simplify square roots. \(\sqrt{\frac{98 r^{5}}{100}}\)
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