Find the cube root of -125.

The cube root of -125 is -5.

To find the cube root of -125, follow these steps:

Step 1: Rewrite -125 as (-1) × (5³).
Step 2: Rewrite -1 as e^(iπ).
Step 3: Rewrite 5³ as (5 × 5 × 5).
Step 4: Apply the property of cube roots: the cube root of a product is equal to the product of the cube roots.
Step 5: Take the cube root of -1, which is e^(iπ/3).
Step 6: Take the cube root of 5 to get 5^(1/3).
Step 7: Combine the cube roots from steps 5 and 6: (-1) × (5^(1/3)) × (e^(iπ/3)).
Step 8: Simplify the expression.
- The cube root of -125 is -5.
- e^(iπ/3) can be simplified to cos(π/3) + i sin(π/3).
- So, the cube root of -125 is -5(cos(π/3) + i sin(π/3)).

Hence, the cube root of -125 is -5(cos(π/3) + i sin(π/3)).

To find the cube root of -125, we can use the following steps:

Step 1: Understand the concept of cube root: The cube root of a number, denoted by ∛x, is the number that when raised to the power of 3 equals x.

Step 2: Calculate the cube root: In the case of -125, the cube root can be found by taking the cube root of the absolute value of -125, and then multiplying it by -1 since the original number is negative.

Therefore, the cube root of -125 can be calculated as follows:

Step 3: Calculate the cube root of the absolute value: The absolute value of -125 is 125. So, we will calculate the cube root of 125.

∛125 = 5

Step 4: Multiply by -1: Since the original number is negative, we multiply the cube root by -1.

∛(-125) = -(∛125) = -5

Therefore, the cube root of -125 is -5.