List three different ways to write 511 as the product of two powers. Explain why all three of your expressions are equal to 511.
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.
One way to write 511 as the product of two powers is:
1) \(511 = 7^1 \times 73^1\)
To understand why this expression equals 511, we can break it down as follows: \(7^1\) means raising 7 to the power of 1, which is simply 7. Similarly, \(73^1\) is equal to 73. Therefore, \(7^1 \times 73^1\) simplifies to \(7 \times 73\), which equals 511.
Another way to express 511 as the product of two powers is:
2) \(511 = 29^1 \times 2^9\)
Again, let's break it down: \(29^1\) is just 29, and \(2^9\) means raising 2 to the power of 9, which is 512. Thus, \(29^1 \times 2^9\) simplifies to \(29 \times 512\), which equals 511.
Finally, we can write 511 as the product of two different powers as:
3) \(511 = 1^9 \times 511^1\)
In this case, \(1^9\) is simply 1, and \(511^1\) is equal to the original number itself, 511. Therefore, \(1^9 \times 511^1\) simplifies to \(1 \times 511\), which equals 511.
All three expressions are equal to 511 because they are different ways of representing the same numerical value. The different powers used in each expression modify the base values (7, 73, 29, 2, 1, and 511) but ultimately result in the same final product of 511.