Problem 35 - Elementary Algebra

Question

Is 9 a perfect cube? Explain.

Step-by-Step Solution

Answer

No, 9 is not a perfect cube.

Step 1: Understand What a Perfect Cube Is

A number is a perfect cube if it can be expressed as the cube of an integer. Mathematically, this means for some integer n, the number can be written as n^3

Step 2: Check Integer Cube Roots

Start testing small integers to see if their cubes equal 9. Begin with: 1^3 = 1 2^3 = 8 3^3 = 27

Step 3: Conclude

Since none of the small integers' cubes result in 9, it confirms that 9 is not a perfect cube as there is no integer n such that n^3 = 9

Key Concepts

Integer Cubes

One important concept to understand is the idea of 'integer cubes'. Simply put, an integer cube is a number that is the result of an integer multiplied by itself three times. This can be written as n^3, where n is an integer.
For example, consider the integers 1, 2, and 3:

• 1^3 = 1 (because 1 * 1 * 1 = 1)
• 2^3 = 8 (because 2 * 2 * 2 = 8)
• 3^3 = 27 (because 3 * 3 * 3 = 27)

These are all perfect cubes because they can be expressed as the cube of an integer. This is a fundamental concept for understanding whether a number is a perfect cube, such as in your original exercise.
If a number cannot be expressed in this way, it is not an integer cube.

Cube Roots

To determine whether a given number is a perfect cube, you can find its integer cube roots. The cube root of a number x is a value y such that y^3 = x. For example, the cube root of 27 is 3 because 3 * 3 * 3 = 27.
When you take the cube root of a number and the result is an integer, the original number is a perfect cube. For instance:

• The cube root of 8 is 2 (since 2^3 = 8)
• The cube root of 64 is 4 (since 4^3 = 64)

If a number like 9 doesn’t have an integer as its cube root, this means 9 is not a perfect cube. You can check this by calculating small integers' cubes until you pass the number in question.

Mathematical Properties

Understanding mathematical properties is essential for solving problems involving cubes and cube roots. One important property is that perfect cubes retain the properties of multiplication over three identical factors. Specifically, for an integer n, n^3 will always result in an integer.
Also, perfect cubes grow very quickly. As you saw from the earlier calculation, the cubes of integers 1, 2, and 3 are 1, 8, and 27, respectively. This rapid growth helps quickly determine if a number is a perfect cube without extensive calculations.
To sum up, checking if a number is a perfect cube involves:

• Finding whether its cube root is an integer.
• Understanding the properties of cubes, like their rapid increase.
• Using these properties simplifies identifying if a number like 9 is a perfect cube or not.