Is 128 a perfect cube? Find its cube root.

To determine if 128 is a perfect cube, 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. In simpler terms, if you can find an integer \( n \) such that \( n^3 = 128 \), then 128 is a perfect cube. Let's break this down step by step.

Finding the Cube Root

The cube root of a number \( x \) is a value \( y \) such that \( y^3 = x \). For 128, we want to find \( y \) such that \( y^3 = 128 \). We can start by estimating the cube root by checking the cubes of integers:

• 1³ = 1
• 2³ = 8
• 3³ = 27
• 4³ = 64
• 5³ = 125
• 6³ = 216

From this list, we can see that 5³ is 125, which is the closest cube to 128, but it is not equal. The next integer, 6, gives us 216, which is too high. Therefore, we can conclude that there is no integer \( n \) such that \( n^3 = 128 \).

Calculating the Exact Cube Root

To find the cube root of 128 more precisely, we can use the formula:

Cube root of \( x \) = \( x^{1/3} \)

So, for 128, we calculate:

Cube root of 128 = \( 128^{1/3} \approx 5.04 \)

This value indicates that while 128 is not a perfect cube, its cube root is approximately 5.04, which is not an integer.

Conclusion on Perfect Cubes

In summary, 128 is not a perfect cube because there is no integer that, when cubed, equals 128. The cube root of 128 is approximately 5.04, confirming that it lies between the cubes of 5 and 6. This understanding of perfect cubes and cube roots is essential in various mathematical contexts, including algebra and number theory.