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Understanding real numbers is fundamental when discussing the domain of any expression. Real numbers include every number that can be found on the number line, encompassing all positive and negative numbers, zero, and every fraction or decimal that falls between these integers. An important characteristic of real numbers is that they do not include imaginary or complex numbers.

When calculating the domain of an expression, particularly in algebra, we must consider which real numbers can be plugged into an equation without causing restrictions like division by zero or taking the square root of a negative number. The domain is essentially the set of all possible real number inputs that the expression can handle without producing undefined results.

Interval notation is a way of documenting a set of numbers between two endpoints in a very concise manner. It is often used to communicate the domain or range of a function. The symbols used in interval notation include parentheses '()', which denote that an endpoint is not included in the set, and brackets '[]', which signal that an endpoint is included.

For example, in the given expression \(\frac{1}{\sqrt[3]{x+4}}\), the domain excludes the value -4, since it would render the expression undefined. Thus, in interval notation, the domain is written as \( (-\infty, -4) \cup (-4, \infty) \), where the union symbol \(\cup\) indicates the combination of two intervals that cover all the real numbers except -4. This notation is an incredibly efficient way to describe continuous sets of numbers without listing each individual element.

An undefined expression in mathematics is a situation where an operation cannot be performed under the rules of arithmetic or algebra. Common instances that lead to undefined expressions include dividing by zero, taking the logarithm of a negative number or zero, or even extracting an even root of a negative number when working within the real numbers.

Returning to our original problem, the expression \(\frac{1}{\sqrt[3]{x+4}}\) can lead to an undefined value when the denominator is zero. Since cube roots can handle negative results, we only need to avoid the denominator equating to zero, which occurs when \(x+4 = 0\). Consequently, \(x\) cannot be -4, leading us to declare the domain as all real numbers except -4, ensuring that the expression is always defined.