The zero exponent

The idea

Any non-zero number raised to the power 0 is 1, that is, x⁰ = 1 for x ≠ 0, defined so that xᵃ ÷ xᵃ = xᵃ⁻ᵃ = x⁰ fits the division rule.

What is 2 to the power 0?

We know what 2^1, 2^2, 2^3, … mean. But what could 2^0 mean? Zero 2s multiplied?

What do you think 2^0 is?

• 0
• 1
• 2

Cancel everything

Make the top and the bottom exponents the same. Then tap every tile to cancel. What is left?

Anything to the power 0 is 1

Any non-zero number raised to the power 0 is 1: x⁰ = 1 for x ≠ 0. It is defined so that xᵃ ÷ xᵃ = xᵃ⁻ᵃ = x⁰ fits the division rule.

2^0 = 2⁴⁻⁴ = 2^4 ÷ 2^4 = (2 × 2 × 2 × 2) ÷ (2 × 2 × 2 × 2) = 1

The same works for every counting number a: 2^a ÷ 2^a = 2ᵃ⁻ᵃ = 2^0 = 1.

Notes

Any non-zero number raised to the power 0 is 1: x⁰ = 1 (x ≠ 0), because xᵃ ÷ xᵃ = xᵃ⁻ᵃ = x⁰.

Check yourself

What is 7^0?

What is 9^0 + 3^0?

Answer: 2

9^0 = 1 and 3^0 = 1, so 9^0 + 3^0 = 1 + 1 = 2.

Why is 2^0 = 1?

What is 2^0 × 3^0 × 4^0?

Answer: 1

Each of 2^0, 3^0 and 4^0 is 1, so the product is 1 × 1 × 1 = 1.

• 0. Not 0. 7^0 = 7¹⁻¹ = 7 ÷ 7 = 1.
• 1 — correct. Yes! Any non-zero number to the power 0 is 1.
• 7. That is 7^1. The exponent here is 0.
• Because 2 multiplied zero times is 0. Zero is not the result. 2^0 is defined so that the division rule still works.
• Because 2^4 ÷ 2^4 = 1, and the division rule makes it 2⁴⁻⁴ = 2^0 — correct. Yes! A number divided by itself is 1, and the rule gives the exponent 4 − 4 = 0.
• Because 0 is the exponent of 1. There is no such rule. The reason is the division rule.