Same exponent: multiply the bases

The idea

mᵃ × nᵃ = (mn)ᵃ: a product of two numbers raised to the same exponent a equals their product raised to that exponent.

The two magical ponds

In one pond the lotuses double every day. In the other they triple every day. Damayanti puts 1 lotus in the doubling pond. After 4 days she moves all of them into the tripling pond for 4 more days.

Doubling pond: 1 × 2 × 2 × 2 × 2 = 2^4. Then the tripling pond: 2^4 × 3 × 3 × 3 × 3 = 2^4 × 3^4

Can 2^4 × 3^4 be written as one power with the exponent 4?

• Yes, as 5^4
• Yes, as 6^4
• No, it cannot

Pair a 2 with a 3

There are four 2s and four 3s. Tap a column to join its 2 and 3.

Multiply the bases

mᵃ × nᵃ = (mn)ᵃ. A product of two numbers raised to the same exponent a equals their product raised to that exponent.

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

The order of the ponds does not matter: 1 × 3^4 × 2^4 = 1 × 2^4 × 3^4. The bases are different, but the exponent must be the same.

Notes

mᵃ × nᵃ = (mn)ᵃ: with the same exponent, multiply the bases and keep the exponent.

Check yourself

What is 2^3 × 5^3 written as one power?

What is 2^3 × 5^3?

Answer: 1000

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

Which one is equal to (3 × 7)^5?

One lotus grows in the doubling pond for 3 days. Then all the lotuses go to the tripling pond for 3 days. How many lotuses are there at the end?

Answer: 216

2^3 × 3^3 = (2 × 3)^3 = 6^3 = 216.

• 7^3. The bases are multiplied (2 × 5), not added.
• 10^3 — correct. Yes! Three pairs of (2 × 5): 2^3 × 5^3 = (2 × 5)^3 = 10^3.
• 10^6. The exponent stays 3. Only the bases join, to make 2 × 5 = 10.
• 3^5 × 7^5 — correct. Yes! The exponent 5 goes with each base: five 3s and five 7s.
• 3^5 × 7. The exponent works on both numbers inside the bracket, not only on the 3.
• 10^5. Do not add the bases. 3 × 7 = 21, so (3 × 7)^5 = 21^5.