1. $2^9 \cdot 31^1$
2. $7^1 \cdot 73^1$
3. $1^1 \cdot 511^1$
All three expressions are equal to 511 because they are all different ways to break 511 down into the product of two powers. The first expression breaks it down into the powers of 2 and 31, the second expression breaks it down into the powers of 7 and 73, and the third expression breaks it down into the powers of 1 and 511. Regardless of which expression is used, the product of the powers will always equal 511.
List three different ways to write 511 as the product of two powers. Explain why all three of your expressions are equal to 511.
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