Negative exponents

The idea

n⁻ᵃ = 1/nᵃ and nᵃ = 1/n⁻ᵃ, where n ≠ 0; a negative exponent arises when the divisor's exponent is larger, and it means the reciprocal of the positive power.

Halve the line five times

A line is 16 units long: 2^4. Halve it 5 times: 2^4 ÷ 2^5.

How long is the line now?

• 0 units
• 1/2 unit
• 1 unit

More 2s at the bottom

Make the bottom exponent bigger than the top. Cancel everything you can. What is left?

Negative exponents are reciprocals

n⁻ᵃ = 1 ÷ nᵃ and nᵃ = 1 ÷ n⁻ᵃ, where n ≠ 0. A negative exponent arises when the divisor’s exponent is larger, and it means the reciprocal of the positive power.

2^4 ÷ 2^5 = (2 × 2 × 2 × 2) ÷ (2 × 2 × 2 × 2 × 2) = 1/2. By the rule: 2⁴⁻⁵ = 2⁻¹, so 2⁻¹ = 1/2.

2^4 ÷ 2^10 = 2⁴⁻¹⁰ = 2⁻⁶ = 1 ÷ 2⁶ = 1/64. 10⁻³ = 1 ÷ 10³ and 7⁻² = 1 ÷ 7².

The other way: 1 ÷ 10⁻³ = 1 ÷ (1 ÷ 10³) = 1 × 10³ = 10³.

Notes

n⁻ᵃ = 1 ÷ nᵃ and nᵃ = 1 ÷ n⁻ᵃ (n ≠ 0): a negative exponent means the reciprocal of the positive power.

Check yourself

What is 3⁻²?

2^4 ÷ 2^10 is 2 to which power? (Type the integer.)

Answer: -6

4 − 10 = −6, so 2^4 ÷ 2^10 = 2⁻⁶ = 1/64.

What is 2⁻³? (Type a fraction, like 1/5.)

Answer: 1/8

2⁻³ = 1 ÷ 2³ = 1 ÷ 8 = 1/8.

How can 7² also be written?

• −9. A negative exponent does not make the number negative. It makes it a reciprocal.
• 1/9 — correct. Yes! 3⁻² = 1 ÷ 3² = 1 ÷ 9.
• −6. That is 3 × (−2). A negative exponent means 1 ÷ 3^2.
• 1 ÷ 7⁻² — correct. Yes! 7⁻² = 1 ÷ 7², so 1 ÷ 7⁻² = 1 ÷ (1 ÷ 7²) = 7².
• 1 ÷ 7². That is 7⁻², the reciprocal of 7².
• −7². A negative sign out front changes the sign. 7² is not the same as −7².