Find the cube root of-512
The cube root of -512 is -8.
To find the cube root of -512, follow these steps:
Step 1: Write down the number -512.
Step 2: Express -512 as a product of prime factors.
-512 = -1 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
Step 3: Group the prime factors into triples.
-512 = (-1) × (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2)
Step 4: Take out one factor from each triple.
-512 = -1 × 2 × 2 × 2 × 2
= -1 × 2^4
Step 5: The cube root of -512 is the cube root of each factor taken separately.
∛(-512) = ∛((-1) × 2^4)
Step 6: The cube root of -1 is -1 and the cube root of 2^4 is 2.
∛(-512) = -1 × 2
= -2
Therefore, the cube root of -512 is -2.
To find the cube root of -512, you can use either a calculator or manual calculation methods. I will explain both methods.
Method 1: Using a Calculator
1. Enter -512 into your calculator.
2. Look for the cube root or "∛" button on your calculator.
3. Press the cube root button and the calculator will give you the result.
Method 2: Manual Calculation
To calculate the cube root manually, follow these steps:
1. Determine the prime factors of the absolute value of the number, 512.
- 512 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 2^9
2. Group the factors into groups of three, starting from the right.
- 2^9 can be grouped as (2 * 2 * 2) * (2 * 2 * 2) * (2 * 2 * 2)
3. The number of groups will determine the sign of the cube root.
- Since there are three groups, the cube root of -512 will be negative.
4. Calculate the cube root of each group.
- The cube root of (2 * 2 * 2) is 2, since 2^3 = 8.
5. Multiply the cube root of each group.
- The product of the cube roots is -2 * -2 * -2 = -8.
6. The cube root of -512 is -8.
Therefore, the cube root of -512 is -8.