Exponents with negative bases

Key Notes :

What is a Negative Base?

• A negative base means the base number is less than zero.
• Example: (−2),(−5),(−10)etc.
• The rules of exponents still apply, but we must be careful with brackets.

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Rules to Remember

Even Exponents make the result positive

• (−a)^even=(+)

• Example: (−3)² = (−3) × (−3) = 9

Odd Exponents keep the result negative

• (−a)^odd=(−)

• Example: (−3)3=(−3)×(−3)×(−3)=−27

Brackets matter!

• (−3)²⁼9( (because both factors are included inside brackets).
• −3²=−9 (because only 3²=9, then negative sign is applied outside).

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3. Examples

Rules to Remember

1. (−4)²=16
2. (−4)³=−64
3. (−2)⁵⁼−32
4. (−2)⁶=64
5. −2⁴=−16

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Key Points

• With brackets → exponent applies to the whole negative number.
• Without brackets → exponent applies only to the positive number; the negative sign stays outside.
• Even powers → always positive.
• Odd powers → always negative.

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Practice Problems

1. (−5)²⁼?
2. (−7)³=?
3. −6² =?
4. (−2⁾⁴=?
5. (−3)⁵=?

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👉 Summary:

When working with negative bases in exponents, always check if brackets are present and whether the power is even or odd.

Learn with an example