List three different ways to write 511 as the product of two powers. Explain why all three of your expressions are equal to 511.
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.
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.
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.
To find three different ways to write 511 as the product of two powers, we can start by considering the prime factorization of 511.
First, we find the prime factors of 511:
511 = 7 * 73
Now, let's look at three different ways to express 511 as the product of two powers:
1. 511 = 7^1 * 73^1:
Here, we express 511 as the product of 7 raised to the power of 1 and 73 raised to the power of 1. Since any number raised to the power of 1 is equal to the number itself, 7^1 = 7 and 73^1 = 73. Multiplying these values gives us 7 * 73 = 511.
2. 511 = 7^9 * 73^0:
In this expression, we express 511 as the product of 7 raised to the power of 9 and 73 raised to the power of 0. Any number raised to the power of 0 equals 1. Therefore, 73^0 = 1. Thus, 7^9 * 73^0 = 7^9 * 1 = 7^9 = 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 = 40353607. However, since we are looking for the product of two powers equal to 511, this expression does not meet our criteria.
3. 511 = 7^0 * 73^1:
In this case, we express 511 as the product of 7 raised to the power of 0 and 73 raised to the power of 1. Any number (except zero) raised to the power of 0 equals 1. Hence, 7^0 = 1. Consequently, 7^0 * 73^1 = 1 * 73 = 73. So, 73 is equal to 511.
In summary, out of the three expressions we considered, two of them satisfy the condition of each being equal to 511: 7^1 * 73^1 and 7^0 * 73^1.