A³ + b³

Definition

The expression a³ + b³ is a special product in algebra, where a and b are variables or constants. It represents the sum of the cubes of two numbers, and is an important concept in factoring special products.

5 Must Know Facts For Your Next Test

1. The expression a³ + b³ can be factored using the formula: a³ + b³ = (a + b)(a² - ab + b²).
2. Factoring a³ + b³ is a special case of factoring the sum of two cubes, which is a common type of factoring problem.
3. The factors of a³ + b³ are (a + b) and (a² - ab + b²), which can be further simplified depending on the values of a and b.
4. If a and b are both positive integers, then a³ + b³ will also be a perfect cube.
5. The expression a³ + b³ has many real-world applications, such as in the calculation of volume and surface area of certain geometric shapes.

Review Questions

• Explain the formula for factoring a³ + b³ and how it is derived.
  • The formula for factoring a³ + b³ is: a³ + b³ = (a + b)(a² - ab + b²). This formula is derived by using the difference of two squares identity: a² - b² = (a + b)(a - b). First, we can rewrite a³ + b³ as (a³) + (b³). Then, we can factor each cube using the difference of two squares identity: a³ = a(a²) and b³ = b(b²). This gives us a(a²) + b(b²) = (a + b)(a² - ab + b²).
• Describe the relationship between a³ + b³ and perfect cubes.
  • If a and b are both positive integers, then a³ + b³ will also be a perfect cube. This is because the sum of two perfect cubes is itself a perfect cube. For example, 1³ + 2³ = 1 + 8 = 9, which is a perfect cube (3³). This relationship is important in understanding the properties of a³ + b³ and how it can be factored and manipulated in algebraic expressions.
• Analyze the real-world applications of the expression a³ + b³ and how it is used in various fields.
  • The expression a³ + b³ has many real-world applications, particularly in the fields of geometry and physics. For example, the volume of a rectangular prism is given by the formula V = length × width × height, which can be written as V = a × b × c, where a, b, and c are the dimensions of the prism. This volume formula can be expanded to V = a³ + b³ + c³, demonstrating the relevance of a³ + b³ in calculating the volume of certain geometric shapes. Additionally, in physics, the expression a³ + b³ can be used to calculate the surface area of spheres and other three-dimensional objects, making it an important concept in fields such as engineering and materials science.

Related terms

Cube: The cube of a number is that number multiplied by itself three times. For example, the cube of 3 is 3 x 3 x 3 = 27.

Perfect Cube: A perfect cube is the cube of an integer. For example, 8 is a perfect cube because it is 2 x 2 x 2.

Difference of Cubes: The difference of two cubes is the expression a³ - b³, which can be factored using a special formula.