Problem 116 - Applied Calculus

Question

For each statement, state in words the values of \(x\) for which each exponential expression is defined. $$ x^{1 / 3} $$

Step-by-Step Solution

Answer

The expression \(x^{1/3}\) is defined for all real numbers \(x\).

Step 1: Understanding Exponential Expressions

The expression given is an exponential form, specifically a root, where the base is \(x\) and the exponent is \(\frac{1}{3}\). This expression represents the cube root of \(x\). An important concept in exponential expressions is that the expression is defined for all real numbers when the exponent is a rational number with an odd denominator.

Step 2: Determine the Domain of the Expression

For the expression \(x^{1/3}\), where the exponent is \(\frac{1}{3}\), it is defined for all real numbers. This is because the cube root of a real number is always defined, regardless of whether the number is positive, negative, or zero. Therefore, \(x\) can be any real number.

Key Concepts

Understanding Cube Roots

A cube root is a special type of root in mathematics. Unlike square roots, which involve finding a number that, when multiplied by itself, gives the original number, the cube root finds a number that, when cubed (multiplied by itself twice), results in the original number. The cube root of a number \(x\) is represented as \(x^{1/3}\). This expression is known as an exponential expression, where the exponent \(\frac{1}{3}\) indicates the cube root.
Cube roots have a unique property that sets them apart from even roots like square roots. They are defined for all real numbers.

• If \(x\) is positive, \(x^{1/3}\) will also be positive.
• If \(x\) is negative, \(x^{1/3}\) will result in a negative real number.
• If \(x\) is zero, then \(x^{1/3}\) equals zero.

This property occurs because when you multiply a negative number by itself an odd number of times, the result remains negative, unlike squaring a negative number, which turns it positive.

Domain of Functions for Exponential Expressions

The term 'domain' refers to the set of all possible values that you can input into a function. It essentially tells you which values are acceptable as inputs for a function to provide a valid output. For the exponential expression \(x^{1/3}\), we are interested in understanding for which values of \(x\) this expression is defined.
The domain for \(x^{1/3}\) is all real numbers. This means you can substitute any real number—positive, negative, or zero—into the expression, and you will end up with a valid real number as a result. This is because unlike square roots, which are undefined for negative numbers in real numbers, cube roots maintain their definition across all real numbers.
When considering domains for other exponential expressions, it is important to look at whether the exponents are fractions with odd or even denominators. Odd denominators, like \(\frac{1}{3}\), ensure the function is defined across the entire set of real numbers, making cube roots highly versatile.

Exploring Real Numbers

Real numbers encompass all the numbers you typically deal with in everyday math. They include positive numbers, negative numbers, zero, fractions, and irrational numbers (numbers that cannot be expressed as fractions). This diverse set forms the basis of most number systems in mathematics.
Real numbers can be divided into various subsets, such as:

• Natural numbers: Counting numbers like 1, 2, 3, ...
• Whole numbers: Natural numbers plus zero.
• Integers: Whole numbers and their negatives, like ..., -3, -2, -1, 0, 1, 2, 3, ...
• Rational numbers: Numbers that can be expressed as fractions.
• Irrational numbers: Numbers like \(\pi\) and \(\sqrt{2}\) that cannot be expressed as a simple fraction.

When dealing with expressions like \(x^{1/3}\), it's crucial to understand that every real number can be a valid input, showcasing the inclusive nature of cube roots. This quality makes real numbers a comprehensive set, allowing exponential expressions like \(x^{1/3}\) to be broadly applicable.