key term - ∛

Definition

The cube root, denoted by the symbol ∛, is a mathematical operation that represents the value which, when multiplied by itself three times, results in the original number. It is a higher root, specifically the third root, that is used to solve equations and problems involving the raising of numbers to the power of one-third.

5 Must Know Facts For Your Next Test

1. The cube root, ∛, is the value that, when multiplied by itself three times, equals the original number.
2. Cube roots can be used to solve equations and problems involving the raising of numbers to the power of one-third, such as finding the side length of a cube with a given volume.
3. Rational exponents, such as 1/3, can be used to represent cube roots (∛) in a more compact and generalized way, allowing for easier calculations and manipulations.
4. Cube roots, like other higher roots, can be used to find the values of perfect cubes, which are numbers that are the cube of an integer.
5. The cube root function is one of the basic higher root functions, along with the square root (√) and other higher roots, and is essential in various mathematical and scientific applications.

Review Questions

• Explain how the cube root, ∛, is related to the concept of raising a number to the power of one-third.
  • The cube root, ∛, is directly related to raising a number to the power of one-third. The cube root of a number is the value that, when multiplied by itself three times, results in the original number. In other words, if we have a number $x$, then $∛x = y$ means that $y^3 = x$. This relationship between the cube root and the one-third power is fundamental to understanding how cube roots can be used to solve equations and problems involving fractional exponents.
• Describe how rational exponents, such as 1/3, can be used to represent cube roots (∛) and the advantages of this notation.
  • Rational exponents, like 1/3, can be used to represent cube roots (∛) in a more compact and generalized way. The expression $x^{1/3}$ is equivalent to $∛x$, as both represent the value that, when raised to the power of 3, equals $x$. Using rational exponents allows for easier manipulation and calculation of cube roots, as well as the ability to extend the concept to higher roots, such as fourth roots (∜) or fifth roots (∛∛). This notation is more versatile and efficient than using the cube root symbol alone, making it a valuable tool in various mathematical and scientific applications.
• Analyze the relationship between cube roots (∛), higher roots, and perfect cubes, and explain how this understanding can be applied to solve problems.
  • Cube roots (∛) are a specific type of higher root, which are mathematical operations that generalize the concept of square roots to finding the value that, when raised to a specific power, equals the original number. The cube root, in particular, finds the value that, when multiplied by itself three times, results in the original number. This relationship between cube roots and perfect cubes (numbers that are the cube of an integer) is essential in solving problems involving the volume of cubes, the side length of cubes, and other applications where the cube root is needed to find the unknown value. Understanding the connections between cube roots, higher roots, and perfect cubes allows you to set up and solve equations more efficiently, as well as recognize patterns and apply the appropriate mathematical concepts to solve a variety of problems.

Related terms

Higher Roots: Higher roots, such as the cube root (∛), fourth root (∜), and so on, are mathematical operations that generalize the concept of square roots to finding the value that, when raised to a specific power, equals the original number.

Rational Exponents:

Rational exponents, which include fractional exponents like 1/3, are a way to represent higher roots, such as the cube root (∛), using exponent notation. This allows for more efficient and generalized calculations involving roots.

Radical Notation: Radical notation, which uses the radical symbol (√), is an alternative way to represent higher roots, such as the cube root (∛). It is often used interchangeably with fractional exponents to express the same mathematical concept.