How many perfect cubes can you find from 1 to 100? How many from -100 to 100?

To determine how many perfect cubes exist within the specified ranges, we first need to understand what a perfect cube is. A perfect cube is a number that can be expressed as the cube of an integer. For example, \(1^3 = 1\), \(2^3 = 8\), and \(3^3 = 27\). Let's break down the calculations for both ranges: from 1 to 100 and from -100 to 100.

Perfect Cubes from 1 to 100

We start by identifying the integers whose cubes fall within the range of 1 to 100. The cubes of integers are calculated as follows:

• 1³ = 1
• 2³ = 8
• 3³ = 27
• 4³ = 64
• 5³ = 125 (exceeds 100)

From this list, we see that the perfect cubes between 1 and 100 are:

• 1
• 8
• 27
• 64

Thus, there are a total of 4 perfect cubes in the range from 1 to 100.

Perfect Cubes from -100 to 100

Next, we expand our range to include negative integers. The cubes of negative integers are also perfect cubes:

• (-1)³ = -1
• (-2)³ = -8
• (-3)³ = -27
• (-4)³ = -64

Now, combining both the positive and negative perfect cubes, we have:

• -64
• -27
• -8
• -1
• 1
• 8
• 27
• 64

In total, we find 8 perfect cubes in the range from -100 to 100.

Summary of Findings

To summarize:

• From 1 to 100, there are 4 perfect cubes: 1, 8, 27, and 64.
• From -100 to 100, there are 8 perfect cubes: -64, -27, -8, -1, 1, 8, 27, and 64.

This exploration of perfect cubes illustrates how both positive and negative integers contribute to the overall count, showcasing the symmetry of mathematical concepts. If you have any further questions or need clarification on any points, feel free to ask!