List two diffrent ways to write 3^7 as the product of two powers
Then explain why both of the expressions are equal to 3^7
1) 3^7 = (3^3) * (3^4)
In this expression, we can break down 3^7 into the product of two powers by adding the exponents. We separate the powers of 3 into 3^3 and 3^4, and when we multiply these two terms together, the exponent rule for multiplication states that we add the exponents. Therefore, (3^3) * (3^4) equals 3^(3+4) = 3^7.
2) 3^7 = (3^2) * (3^5)
In this expression, we can also break down 3^7 into the product of two powers by grouping the exponents differently. Here, we have 3^7 represented as (3^2) * (3^5). Again, using the exponent rule for multiplication, we add the exponents and obtain 3^(2+5) = 3^7.
Both of these expressions are equal to 3^7 because they both follow the exponent rule for multiplication, which states that when we multiply two powers with the same base, we add their exponents. Therefore, regardless of how we break down 3^7 into the product of two powers, the result is still 3^7.