A power of a power: multiply the exponents

The idea

(nᵃ)ᵇ = (nᵇ)ᵃ = nᵃᵇ: when a power of n is raised to a power, the exponents a and b are multiplied.

4 to the power 6, in two ways

(4 × 4 × 4) × (4 × 4 × 4) is 4^3 × 4^3, the square of 4^3. We write it (4^3)^2.

(4^3)^2 means 4^3 × 4^3. How many 4s are multiplied in all?

• 5
• 6
• 9

Group the grid

Each tile is a factor. Group the grid by rows or by columns. Count the tiles.

Multiply the exponents

(nᵃ)ᵇ = (nᵇ)ᵃ = nᵃᵇ. When a power of n is raised to a power, the exponents a and b are multiplied.

(4 × 4 × 4) × (4 × 4 × 4) = 4^3 × 4^3 = (4^3)^2 = 64 × 64 = 4096

(4 × 4) × (4 × 4) × (4 × 4) = 4^2 × 4^2 × 4^2 = (4^2)^3 = 16 × 16 × 16 = 4096

2^10 = (2^2)^5 = (2^5)^2, because 2 × 5 = 5 × 2 = 10.

Notes

(nᵃ)ᵇ = (nᵇ)ᵃ = nᵃᵇ: when a power of n is raised to a power, the exponents are multiplied.

Check yourself

Write (2^3)^2 as one power of 2.

(5^2)^3 is the same as 5 raised to which power?

Answer: 6

3 groups of 2 fives each: 2 × 3 = 6, so (5^2)^3 = 5^6.

2^10 = (2^5) raised to which power?

Answer: 2

5 × 2 = 10, so 2^10 = (2^5)^2.

(4^2)^3 and (4^3)^2. Which one is bigger?

• 2^5. That adds 3 + 2. Add only when powers are multiplied. Here 2^3 is repeated 2 times: 3 twos + 3 twos.
• 2^6 — correct. Yes! 2^3 × 2^3 has 3 + 3 = 6 twos, which is 3 × 2.
• 2^9. That is 3^2. The outer 2 only says how many times 2^3 is used.
• (4^2)^3. It is 4^6 = 4096. But so is the other one.
• (4^3)^2. It is 4^6 = 4096. But so is the other one.
• They are equal — correct. Yes! 2 × 3 = 3 × 2 = 6, so both are 4^6 = 4096.