Cubes of Negative Numbers
When you cube a negative number, the result is always negative. This is a direct consequence of how multiplication of negative numbers works.
Recall that multiplying two negative numbers gives a positive result, but multiplying three negative numbers gives a negative result. Since cubing means multiplying a number by itself three times, (−n)³ = (−n) × (−n) × (−n), the answer is always negative.
This property distinguishes cubes from squares. While the square of any number (positive or negative) is always positive, the cube preserves the sign of the original number. Understanding this is important when solving equations involving cubes and cube roots.
What is Cubes of Negative Numbers?
Definition: The cube of a negative number −n is:
(−n)³ = −n × (−n) × (−n) = −n³
Where:
- −n = a negative number
- (−n)³ = the cube of that negative number
- The result is always negative
Sign rule for cubes:
- Positive × Positive × Positive = Positive
- Negative × Negative × Negative = Negative
Step-by-step sign analysis:
- (−n) × (−n) = +n² (negative × negative = positive)
- (+n²) × (−n) = −n³ (positive × negative = negative)
Methods
Method 1: Direct multiplication
- Multiply the first two factors: (−n) × (−n) = +n²
- Multiply the result by (−n): n² × (−n) = −n³
- The cube is −n³
Method 2: Cube the absolute value and attach the sign
- Find the absolute value of the number (ignore the sign).
- Cube the absolute value.
- Attach a negative sign to the result.
Example: Find (−6)³
- Absolute value of −6 is 6
- 6³ = 216
- Attach negative sign: (−6)³ = −216
Method 3: Using the identity (−a)³ = −a³
- This identity shows that the cube of a negative number equals the negative of the cube of its absolute value.
- (−4)³ = −(4³) = −64
Solved Examples
Example 1: Example 1: Cube of a small negative number
Problem: Find (−3)³.
Solution:
- (−3)³ = (−3) × (−3) × (−3)
- = 9 × (−3)
- = −27
Answer: (−3)³ = −27.
Example 2: Example 2: Cube of −7
Problem: Find the cube of −7.
Solution:
- (−7)³ = (−7) × (−7) × (−7)
- = 49 × (−7)
- = −343
Answer: (−7)³ = −343.
Example 3: Example 3: Cube of a negative fraction
Problem: Find (−2/3)³.
Solution:
Using (−a/b)³ = −a³/b³:
- (−2/3)³ = −(2³)/(3³)
- = −8/27
Answer: (−2/3)³ = −8/27.
Example 4: Example 4: Checking a perfect cube
Problem: Is −216 a perfect cube?
Solution:
- Check if 216 is a perfect cube: 216 = 6³
- Therefore −216 = (−6)³
- Yes, −216 is a perfect cube.
Answer: Yes, −216 = (−6)³.
Example 5: Example 5: Cube of a negative decimal
Problem: Find (−0.5)³.
Solution:
- Convert to fraction: −0.5 = −1/2
- (−1/2)³ = −(1³)/(2³) = −1/8
- = −0.125
Answer: (−0.5)³ = −0.125.
Example 6: Example 6: Comparing (−4)² and (−4)³
Problem: Find (−4)² and (−4)³. Compare the results.
Solution:
- (−4)² = (−4) × (−4) = 16 (positive)
- (−4)³ = (−4) × (−4) × (−4) = 16 × (−4) = −64 (negative)
The square is positive but the cube is negative.
Answer: (−4)² = 16 and (−4)³ = −64.
Example 7: Example 7: Finding a number from its cube
Problem: If x³ = −512, find x.
Solution:
- Since the cube is negative, x must be negative.
- Find the cube root of 512: 512 = 8³
- So x³ = −512 = (−8)³
- x = −8
Answer: x = −8.
Example 8: Example 8: Product of two negative cubes
Problem: Find (−2)³ × (−3)³.
Solution:
- (−2)³ = −8
- (−3)³ = −27
- (−8) × (−27) = 216
Alternatively: (−2)³ × (−3)³ = [(−2)(−3)]³ = 6³ = 216
Answer: (−2)³ × (−3)³ = 216.
Example 9: Example 9: Sum of cubes
Problem: Find (−5)³ + (3)³.
Solution:
- (−5)³ = −125
- (3)³ = 27
- −125 + 27 = −98
Answer: (−5)³ + (3)³ = −98.
Example 10: Example 10: Cube of −1
Problem: Show that (−1)³ = −1 and verify for (−1)⁵.
Solution:
- (−1)³ = (−1) × (−1) × (−1) = 1 × (−1) = −1
- (−1)⁵ = (−1)³ × (−1)² = (−1)(1) = −1
Pattern: (−1) raised to any odd power equals −1.
Answer: (−1)³ = −1 and (−1)⁵ = −1.
Real-World Applications
Real-world applications:
- Temperature changes: If a substance cools at a rate proportional to the cube of a factor, negative cubes model decreasing temperatures below zero.
- Physics — vectors: Forces in the opposite direction can involve negative cubes in three-dimensional calculations.
- Coordinate geometry: Cubing negative coordinates is needed when working with cubic curves and 3D graphs.
- Algebra: Solving cubic equations often requires finding cubes of negative numbers.
- Volume problems: While physical volume is positive, mathematical problems may involve negative cubes in equations.
Key Points to Remember
- The cube of a negative number is always negative.
- (−n)³ = −n³ for all values of n.
- Step-by-step: (−n) × (−n) = +n², then (+n²) × (−n) = −n³.
- Cubes preserve the sign of the original number (unlike squares).
- To cube a negative number, cube the absolute value and make it negative.
- A negative perfect cube is one that equals (−n)³ for some natural number n.
- The cube root of a negative number is negative: ∛(−a³) = −a.
- (−1) raised to any odd power equals −1.
- (−a)³ × (−b)³ = (ab)³ (product of two negative cubes is positive).
- If x³ is negative, then x must be negative.
Practice Problems
- Find (−9)³.
- Find (−3/4)³.
- Is −1000 a perfect cube? If yes, find the number.
- Find (−2)³ + (−3)³.
- If x³ = −343, find x.
- Compare (−6)² and (−6)³.
- Find (−0.2)³.
- Simplify: (−4)³ ÷ (−2)³.
Frequently Asked Questions
Q1. Is the cube of a negative number always negative?
Yes. The cube of any negative number is always negative. (−n)³ = −n³.
Q2. Why is the cube of a negative number negative while the square is positive?
Squaring involves two multiplications: (−n) × (−n) = +n² (negative × negative = positive). Cubing has one more step: (+n²) × (−n) = −n³ (positive × negative = negative).
Q3. What is (−1)³?
(−1)³ = (−1) × (−1) × (−1) = −1. The cube of −1 is −1.
Q4. Can a negative number be a perfect cube?
Yes. A negative number is a perfect cube if its absolute value is a perfect cube. For example, −64 = (−4)³ is a perfect cube.
Q5. What is the cube root of a negative number?
The cube root of a negative number is negative. ∛(−27) = −3 because (−3)³ = −27.
Q6. How do you cube a negative fraction?
Cube the numerator and denominator separately, then attach a negative sign. (−a/b)³ = −a³/b³. Example: (−2/5)³ = −8/125.
Q7. Is (−a)³ the same as −a³?
Yes. (−a)³ = (−a)(−a)(−a) = −a³. They give the same result.
Q8. What happens when you multiply two negative cubes?
The product of two negative cubes is positive. (−a)³ × (−b)³ = (−a³)(−b³) = a³b³ = (ab)³.
