Join the powers: add the exponents

The idea

nᵃ × nᵇ = nᵃ⁺ᵇ: when powers of the same base n are multiplied, their exponents a and b are added.

The stones that shine

Three daughters each got 3 baskets. Each basket had 3 keys. Each key opens 3 rooms. Rooms: 3 × 3 × 3 × 3 = 3^4 = 81.

Each room has 3 tables. Each table has 3 necklaces. Each necklace has 3 diamonds.

We know 3^4 = 81 rooms. How can we find the total number of diamonds?

• Add 81 and 27
• Multiply 81 by 27
• Count all the diamonds from the start

Join two groups of threes

Make two groups of 3s. Press + and −. The tiles join into one row.

Add the exponents

nᵃ × nᵇ = nᵃ⁺ᵇ. When powers of the same base n are multiplied, their exponents a and b are added.

Diamonds: 3 × 3 × 3 × 3 × 3 × 3 × 3 = 3^7 3^7 = (3 × 3 × 3 × 3) × (3 × 3 × 3) = 3^4 × 3^3 = 81 × 27 = 2187

The same works with letters: p^4 × p^6 = (p × p × p × p) × (p × p × p × p × p × p) = p^10

Notes

nᵃ × nᵇ = nᵃ⁺ᵇ: when powers of the same base are multiplied, the exponents are added.

Check yourself

What is 3^2 × 3^5?

p^4 × p^6 is p to which power?

Answer: 10

Four p’s and six p’s make ten p’s: p^4 × p^6 = p^10.

Use 3^4 = 81 and 3^3 = 27 to find 3^7.

Answer: 2187

3^7 = 3^4 × 3^3 = 81 × 27 = 2187.

Which one can be written as one power by adding the exponents?

• 3^10. That multiplies the exponents: 2 × 5. Count the 3s instead: 2 + 5.
• 3^7 — correct. Yes! Two 3s and five 3s make seven 3s: 2 + 5 = 7, so 3^7.
• 9^7. The base stays 3. The multiplication only joins the 3s.
• 5^3 × 5^4 — correct. Yes! The base 5 is the same: 5^3 × 5^4 = 5^7.
• 5^3 × 2^4. The bases are different (5 and 2), so there are no matching factors to count together.
• 5^3 + 5^4. This is an addition, not a multiplication. Adding exponents works only when powers are multiplied.