Same exponent: divide the bases

The idea

mᵃ/nᵃ = (m/n)ᵃ, where n ≠ 0: a quotient of two numbers raised to the same exponent equals their quotient raised to that exponent.

Four sixes, four twos

Look at 6^4 ÷ 2^4: (6 × 6 × 6 × 6) ÷ (2 × 2 × 2 × 2). Both parts have 4 factors.

Is 6^4 ÷ 2^4 equal to 3^4?

• Yes, it is 3^4
• No, it is bigger than 3^4
• No, it is smaller than 3^4

Divide in pairs

Tap a column to divide its 6 by its 2 (the line means ÷).

Divide the bases

mᵃ ÷ nᵃ = (m ÷ n)ᵃ, where n ≠ 0. A quotient of two numbers raised to the same exponent equals their quotient raised to that exponent.

6^4 ÷ 2^4 = (6 × 6 × 6 × 6) ÷ (2 × 2 × 2 × 2) = (6 ÷ 2) × (6 ÷ 2) × (6 ÷ 2) × (6 ÷ 2) = (6 ÷ 2)^4 = 3^4 = 81

Check: 6^4 = 1296 and 2^4 = 16, and 1296 ÷ 16 = 81.

Notes

mᵃ ÷ nᵃ = (m ÷ n)ᵃ: with the same exponent, divide the bases and keep the exponent (n ≠ 0).

Check yourself

What is 10^3 ÷ 5^3 written as one power?

What is 10^3 ÷ 5^3?

Answer: 8

10^3 ÷ 5^3 = (10 ÷ 5)^3 = 2^3 = 8. Check: 1000 ÷ 125 = 8.

Which one is equal to (12 ÷ 4)^5?

What is 12^2 ÷ 3^2?

Answer: 16

12^2 ÷ 3^2 = (12 ÷ 3)^2 = 4^2 = 16. Check: 144 ÷ 9 = 16.

• 5^3. That subtracts the bases (10 − 5). We divide them: 10 ÷ 5 = 2.
• 2^3 — correct. Yes! 10 ÷ 5 = 2 and the exponent stays 3: 10^3 ÷ 5^3 = (10 ÷ 5)^3 = 2^3.
• 2^1. The exponent does not change. It is still 3: only the bases are divided.
• 12^5 ÷ 4^5 — correct. Yes! The exponent 5 goes with both numbers: 12^5 ÷ 4^5.
• 12^5 ÷ 4. The exponent works on both numbers, not only on the 12.
• 8^5. Do not subtract the bases. 12 ÷ 4 = 3, so this is 3^5.