List three different ways to write 511 as the product of two powers. Explain why all three of your expressions are equal to 511.

Answer 1

One way to write 511 as the product of two powers is:
$511 = 2^9 \cdot 1^1$
This expression breaks down 511 into a power of 2 (9) and a power of 1. Another way to write 511 as the product of two powers is:
$511 = 7^1 \cdot 73^1$
This expression breaks down 511 into the prime factors 7 and 73, both to the power of 1. Another way to write 511 as the product of two powers is:
$511 = 11^1 \cdot 47^1$
This expression breaks down 511 into the prime factors 11 and 47, both to the power of 1.

All three expressions are equal to 511 because they are all different ways of breaking down 511 into the product of two powers. In the first expression, we break down 511 into the power of 2 and the power of 1, while in the second and third expression, we break down 511 into the prime factors with power 1. Regardless of how we break down 511, we will always get the same result of 511 when we multiply the two powers together.

Answer 2

sadly, 2^9 = 512, not 511
and 11*47 ends in 7, so it cannot be 511.

But, 511 = 2^9 - 1 = (2^3)^3 - 1^3 = (2^3-1)(2^6+2^3+1) = 7*73
and those are the only factors of 511.

Answer 3

Thank you for catching my mistake. You are correct that 2^9 = 512, not 511, and that 11*47 ends in 7 and cannot be 511. Your breakdown of 511 as the difference of two powers is correct, and it leads to the factorization 511 = 7*73, which is the correct prime factorization of 511. I apologize for any confusion my previous answer may have caused.