Divide powers: subtract the exponents

The idea

nᵃ ÷ nᵇ = nᵃ⁻ᵇ, where n ≠ 0 and a > b: when powers of the same base are divided, the exponent of the divisor is subtracted.

Erase half of a line

A line is 16 units long. That is 2^4 units. Erase half of it: 2^4 ÷ 2 = 8 units. Erase half again: 4 units.

The line is halved 3 times: 2^4 ÷ 2^3. How long is it now?

• 1 unit
• 2 units
• 8 units

Cancel the 2s

Tap a tile. It cancels with a tile in the other row. Cancel everything you can. What is left?

Subtract the exponents

nᵃ ÷ nᵇ = nᵃ⁻ᵇ, where n ≠ 0 and a > b. When powers of the same base are divided, the exponent of the divisor is subtracted.

2^4 ÷ 2^3 = (2 × 2 × 2 × 2) ÷ (2 × 2 × 2) = 2 = 2⁴⁻³ = 2¹

Halving 16 cm twice: 2^4 ÷ 2^2 = (2 × 2 × 2 × 2) ÷ (2 × 2) = 2 × 2 = 2^2 = 4 units.

Notes

nᵃ ÷ nᵇ = nᵃ⁻ᵇ (n ≠ 0, a > b): dividing powers of the same base subtracts the exponent of the divisor.

Check yourself

What is 5^7 ÷ 5^3 written as one power?

3^8 ÷ 3^5 is 3 to which power?

Answer: 3

8 − 5 = 3, so 3^8 ÷ 3^5 = 3^3.

What is 2^6 ÷ 2^2?

Answer: 16

2^6 ÷ 2^2 = 2^4 = 16. Check: 64 ÷ 4 = 16.

4^7 = 16384 and 4^5 = 1024. How many times larger is 4^7 than 4^5?

Answer: 16

4^7 ÷ 4^5 = 4^2 = 16, so 4^7 is 16 times larger.

• 5^10. That adds 7 + 3. We add exponents when we multiply powers. For division we subtract.
• 5^4 — correct. Yes! 7 − 3 = 4: four 5s are left after cancelling three.
• 1^4. The base does not change. It stays 5; only the exponents are subtracted.