Cube Root of -64: Finding the Solution

In the realm of mathematics, understanding the concept of roots is crucial. While we often encounter square roots, cube roots are equally important, especially when dealing with volumes and three-dimensional objects.

What is a Cube Root?

A cube root of a number is a value that, when multiplied by itself three times, results in the original number. In simpler terms, it’s the inverse operation of cubing a number.

For example, the cube root of 8 is 2, because 2 x 2 x 2 = 8.

Finding the Cube Root of -64

To find the cube root of -64, we need to determine the number that, when multiplied by itself three times, equals -64.

Here’s how we can find the solution:

• **Think about the factors of -64:** -64 is a negative number, so the cube root will also be negative. The prime factorization of -64 is -1 x 2 x 2 x 2 x 2 x 2 x 2.
• **Group the factors:** We can group the factors into three sets of -2: (-2) x (-2) x (-2) = -8
• **Calculate the cube root:** Since (-2) x (-2) x (-2) = -64, the cube root of -64 is -4.

Key Points

• The cube root of a negative number is always negative.
• The cube root of a positive number is always positive.
• Cube roots can be found using prime factorization and grouping factors.

Examples

Here are some additional examples:

• The cube root of 27 is 3, because 3 x 3 x 3 = 27.
• The cube root of -125 is -5, because (-5) x (-5) x (-5) = -125.

Conclusion

Understanding cube roots is essential for various mathematical concepts and applications, especially when dealing with volumes and three-dimensional geometry. By applying the principles of prime factorization and grouping factors, we can efficiently determine the cube root of any number.