list two different ways to write 3^7 as a product of two powers
Way 1:
Way 2:
Explain why both of your expressions are equal to 3^7
Way 1: 3^7 = (3^6) * 3 = (3^2)^3 * 3 = 9^3 * 3
Way 2: 3^7 = (3^3) * (3^4) = 27 * (3^4)
Both of these expressions are equal to 3^7 because they follow the exponentiation rule that states when multiplying two powers with the same base, you add the exponents.
In Way 1, we first rewrite 3^7 as (3^6) * 3. Then, we apply the rule that (a^b)^c = a^(b*c) to rewrite (3^6) as (3^2)^3 because 6 is equal to 2*3. Finally, we simplify by calculating 9^3 and obtain 729. So, (3^6) * 3 is equal to 9^3 * 3, which is equal to 3^7.
In Way 2, we separate 3^7 as (3^3) * (3^4). Then, we use the rule of multiplying powers with the same base to rewrite this as 27 * (3^4). By evaluating 3^4, we get 81. Therefore, (3^3) * (3^4) is equal to 27 * (3^4), which is equal to 3^7.