Powers with Negative Base
When a negative number is raised to a power, the sign of the result depends on whether the exponent is even or odd. Two negative numbers multiplied give a positive result, so even powers of negative numbers are always positive.
Understanding this pattern is essential for simplifying expressions with negative bases in algebra and higher mathematics.
What is Powers with Negative Base - Grade 7 Maths (Exponents and Powers)?
Definition: A power with a negative base is written as (−2)n. The sign of the result follows a simple rule:
- If n is even: (−a)n = positive
- If n is odd: (−a)n = negative
Powers with Negative Base Formula
Sign Rule:
(−a)even = +aeven | (−a)odd = −aodd
Important distinction:
- (−3)2 = (−3) × (−3) = 9 (base is −3)
- −32 = −(3 × 3) = −9 (only 3 is squared)
Types and Properties
Pattern of signs:
- (−2)1 = −2 (odd → negative)
- (−2)2 = 4 (even → positive)
- (−2)3 = −8 (odd → negative)
- (−2)4 = 16 (even → positive)
Special case: (−1)n = 1 if n is even, −1 if n is odd.
Solved Examples
Example 1: Even Exponent
Problem: Find (−4)2.
Solution:
- Exponent 2 is even → result is positive.
- (−4)2 = (−4) × (−4) = 16
Answer: 16.
Example 2: Odd Exponent
Problem: Find (−3)3.
Solution:
- Exponent 3 is odd → result is negative.
- (−3)3 = (−3) × (−3) × (−3) = 9 × (−3) = −27
Answer: −27.
Example 3: Brackets vs No Brackets
Problem: Find (−5)2 and −52. Are they equal?
Solution:
- (−5)2 = (−5) × (−5) = 25
- −52 = −(5 × 5) = −25
Answer: They are not equal. Brackets matter.
Example 4: Powers of (−1)
Problem: Find (−1)50 and (−1)75.
Solution:
- 50 is even → (−1)50 = 1
- 75 is odd → (−1)75 = −1
Answer: (−1)50 = 1, (−1)75 = −1.
Real-World Applications
Real-world uses:
- Algebra: Evaluating expressions like (−x)4 when substituting values.
- Sequences: Alternating sign patterns such as 1, −2, 4, −8, 16, ... follow (−2)n.
- Temperature: Repeated drops in temperature can be modelled with negative base concepts.
Key Points to Remember
- (−a)even is always positive.
- (−a)odd is always negative.
- (−a)n ≠ −an when n is even.
- (−1)n = 1 (even) or −1 (odd).
- (−a)0 = 1 for any non-zero a.
- Laws of exponents apply to negative bases too.
Practice Problems
- Find (−5)³.
- Find (−2)⁷.
- Is (−6)² equal to −6²?
- Which is greater: (−4)³ or (−4)²?
Frequently Asked Questions
Q1. What happens when a negative number is raised to an even power?
The result is always positive. For example, (−3)² = 9. An even number of negative factors multiply to give a positive product.
Q2. Is (−3)² the same as −3²?
No. (−3)² = 9 (base is −3). −3² = −9 (only 3 is squared, then the negative is applied). Brackets matter.
Q3. What is (−1) raised to any power?
(−1) to an even power = 1. (−1) to an odd power = −1. The absolute value is always 1; only the sign alternates.
