Cube Root

Definition

The cube root is a mathematical operation that finds the value that, when multiplied by itself three times, equals a given number. It is a way of reversing the process of raising a number to the power of three, and is represented by the symbol $\sqrt[3]{}$.

5 Must Know Facts For Your Next Test

1. The cube root of a number is the value that, when multiplied by itself three times, equals the original number.
2. Cube roots can be used to find the side length of a cube with a given volume.
3. Cube roots of perfect cubes are always integers, while cube roots of non-perfect cubes are irrational numbers.
4. The cube root of a negative number is the negative of the cube root of the absolute value of that number.
5. Cube roots are often used in scientific and engineering applications, such as calculating the volume of a cube or the intensity of a sound wave.

Review Questions

• Explain how the cube root is related to the concept of rational and irrational numbers.
  • The cube root is closely related to the concepts of rational and irrational numbers. The cube root of a perfect cube, such as 8 or 27, is a rational number because it can be expressed as the ratio of two integers (e.g., $\sqrt[3]{8} = 2$). However, the cube root of a non-perfect cube, such as 10 or $\sqrt{2}$, is an irrational number because it cannot be expressed as the ratio of two integers. This is because the process of raising a number to the power of three and then finding the cube root is not always reversible with rational numbers, leading to the existence of irrational numbers.
• Describe how the cube root can be used to find the side length of a cube with a given volume.
  • The cube root can be used to determine the side length of a cube given its volume. The formula for the volume of a cube is $V = s^3$, where $s$ is the side length of the cube. To find the side length, we can take the cube root of the volume: $s = \sqrt[3]{V}$. For example, if a cube has a volume of 27 cubic units, the side length can be found by taking the cube root: $s = \sqrt[3]{27} = 3$ units. This relationship between the cube root and the side length of a cube is an important application of the cube root operation.
• Analyze the properties of cube roots and how they differ from the properties of square roots.
  • The cube root has several properties that differentiate it from the square root. Unlike square roots, which always result in a positive value, cube roots can be positive or negative depending on the original number. For example, the cube root of 8 is 2, but the cube root of -8 is -2. Additionally, the cube root of a perfect cube is always an integer, while the cube root of a non-perfect cube is an irrational number. This is in contrast to square roots, where the square root of a perfect square is an integer, but the square root of a non-perfect square is an irrational number. These differences in the properties of cube roots and square roots are important to understand when working with these mathematical operations in the context of rational and irrational numbers.

Related terms

Rational Number:

A rational number is a number that can be expressed as a ratio of two integers, such as $\frac{1}{2}$ or $-3$.

Irrational Number:

An irrational number is a number that cannot be expressed as a ratio of two integers, such as $\pi$ or $\sqrt{2}$.

Perfect Cube: A perfect cube is a number that is the cube of an integer, such as $8 = 2^3$ or $27 = 3^3$.