Fourth Root

Definition

The fourth root of a number is the value that, when raised to the power of four, equals the original number. It is denoted by the symbol $\sqrt[4]{x}$ and represents the inverse operation of raising a number to the fourth power.

5 Must Know Facts For Your Next Test

1. The fourth root of a number is the value that, when multiplied by itself four times, equals the original number.
2. The fourth root of a number is often used to find the side length of a cube with a given volume.
3. The fourth root can be used to find the domain and range of functions that involve fourth roots, such as $f(x) = \sqrt[4]{x}$.
4. The fourth root is a special case of the general root function, where the index is 4 instead of the more common square root (index 2) or cube root (index 3).
5. The fourth root is a monotonically increasing function, meaning that as the input increases, the output also increases.

Review Questions

• Explain how the fourth root is related to the domain and range of a function.
  • The fourth root is closely related to the domain and range of functions that involve fourth roots. The domain of a function like $f(x) = \sqrt[4]{x}$ is typically the set of non-negative real numbers, as the fourth root is only defined for non-negative inputs. The range of such a function is also the set of non-negative real numbers, as the fourth root will always produce a non-negative output. Understanding the properties of the fourth root, such as its monotonically increasing behavior, is crucial for determining the domain and range of functions that include fourth roots.
• Describe the relationship between the fourth root and the fourth power of a number.
  • The fourth root and the fourth power of a number are inverse operations. If $x$ is a number, then $\sqrt[4]{x^4} = x$, and $x^4 = \sqrt[4]{x}^4$. This means that the fourth root undoes the fourth power, and vice versa. This relationship is important for understanding the properties of fourth roots and how they can be used to solve equations or simplify expressions involving fourth powers.
• Analyze the behavior of the fourth root function and how it differs from other root functions.
  • The fourth root function, $f(x) = \sqrt[4]{x}$, has unique properties that distinguish it from other root functions, such as the square root or cube root. Unlike the square root, which is only defined for non-negative numbers, the fourth root is defined for all real numbers, both positive and negative. Additionally, the fourth root function is monotonically increasing, meaning that as the input increases, the output also increases. This behavior contrasts with the square root function, which is not monotonically increasing. Understanding these differences in behavior is crucial for analyzing the domain, range, and transformations of functions involving fourth roots.

Related terms

Radical: A symbol (√) used to represent the root of a number, such as the square root, cube root, or fourth root.

Exponent:

The number that indicates how many times a base number is to be multiplied by itself.

Power: The result of raising a number to an exponent, such as $x^4$ which is the fourth power of $x$.