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Chapter 10: Problem 15

Use radical notation to write each expression. Simplify if possible. $$ -16^{1 / 4} $$

Short Answer

Expert verified
The expression \(-16^{1/4}\) is \( \sqrt[4]{-16} \), which is not a real number.

Step by step solution

01

Understand Radical Notation

Radical notation involves using a root symbol, \( \sqrt{ } \), to represent a number that is being raised to a fractional power. The expression \( -16^{1/4} \) means we need to find the fourth root of \(-16\).
02

Apply Radical Notation

Rewrite the expression \(-16^{1/4}\) using radical notation. The fourth root (or quarter power) of a number \( x \) is written as \( \sqrt[4]{x} \). Thus, \(-16^{1/4}\) can be rewritten as \( \sqrt[4]{-16} \).
03

Simplify if Possible

Determine if the expression can be simplified. However, it's important to note that \(-16\) is a negative number, and we are searching for a real number that is the fourth root of a negative number, which does not exist in the real number system. Hence, \( \sqrt[4]{-16} \) cannot be simplified to a real number.

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Key Concepts

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

Fractional Exponents
Fractional exponents are a powerful mathematical tool used to express roots in a more generalized form. They connect the dots between exponentiation and root operations.
Understanding how to interpret and manipulate them is crucial:
  • When you see an exponent like \( \frac{1}{n} \), it indicates the \( n \)-th root of a number.
  • For example, \( x^{1/2} \) is the square root of \( x \), and \( x^{1/3} \) represents the cubic root.
  • More generally, \( x^{m/n} \) translates to taking the \( n \)-th root of \( x \), then raising the result to the \( m \)-th power.
These operations allow us to perform root calculations using exponent rules, giving us a seamless way to shift between radical notation and exponential notation. This concept helps simplify expressions and perform operations like multiplication and division more easily with roots.
Root Symbol
The root symbol, often seen as \( \sqrt{} \), is commonly used in mathematics to represent square roots, but it can also express other roots:
  • To denote a square root, the basic symbol \( \sqrt{x} \) is used.
  • For higher roots, we extend this by indicating the degree of the root, such as \( \sqrt[3]{x} \) for cube roots, \( \sqrt[4]{x} \) for fourth roots, and so on.
  • The number outside the radical is called the index, and it informs us about the degree of the root.
Radical notation is intuitive and visually signifies the division or splitting process underlying root calculation. It's important to remember that while square roots are straightforward for positive numbers, calculating higher-order roots of negative numbers may not always yield a result within the real number system.
Real Number System
The real number system is a key concept in understanding where numbers and their operations fit in our everyday math understanding. It encompasses all possible number values that can be represented on a continuous number line:
  • Includes integers, rational numbers (like fractions), and irrational numbers (numbers that cannot be expressed as simple fractions, like \( \pi \) and \( \sqrt{2} \)).
  • All these numbers can be laid out on a single line, called the real number line, where each point corresponds to a real number value.
One major aspect of the real number system is that it does not include the roots of negative numbers when looking for even roots, such as the square or fourth root. Since such a root doesn't exist in the real numbers, it's crucial to recognize the limitations of calculating roots of negative numbers within this system.

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Most popular questions from this chapter

The Mosteller formula for calculating adult body surface area is \(B=\sqrt{\frac{h w}{3131}},\) where \(B\) is an individual's body surface area in square meters, \(h\) is the individual's height in inches, and \(w\) is the individual's weight in pounds. Use this information to answer Exercises 109 and \(110 .\) Round answers to 2 decimal places. Find the body surface area of an individual who is 74 inches tall and who weighs 225 pounds.

Identify the domain and then graph each function. $$f(x)=\sqrt{x}+2$$

Write each integer as a product of two integers such that one of the factors is a perfect cube. For example, write 24 as \(8 \cdot 3,\) because 8 is a perfect cube. $$ 54 $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[12]{a^{8} b^{4}} $$

Use a calculator to write a four-decimal-place approximation of each number. $$ 18^{3 / 5} $$
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