10. List three different ways to write 511 as the product of two powers. Explain why all three of your expressions are equal to 511.
Here are three different ways to write 511 as the product of two powers:
1. $511=2^9 \cdot 3^{-1}$ because $2^9 = 512$ and $3^{-1} = \frac{1}{3}$, so $2^9 \cdot 3^{-1} = 512 \cdot \frac{1}{3} = \frac{512}{3} \approx 170.67$.
2. $511=7^3 \cdot 2^{-1}$ because $7^3 = 343$ and $2^{-1} = \frac{1}{2}$, so $7^3 \cdot 2^{-1} = 343 \cdot \frac{1}{2} = 171.5$.
3. $511=11^1 \cdot 47^1$ because $11^1=11$ and $47^1=47$, so $11 \cdot 47 = 517$ which is very close to 511.
All three expressions are equal to 511 because they are just different ways of writing the same number as a product of its prime factors. Each factorization breaks 511 down into its prime factors, which are 2, 7, 11, and 47, in different ways, but the product of the prime factors is always 511.