What are three distinct methods of expressing 1,024 as the product of two powers, and how can you demonstrate that all three of these expressions are equivalent to 1,024?
To express 1,024 as the product of two powers, we need to find pairs of powers that multiply together to equal 1,024. Let's explore three different methods for expressing 1,024 as the product of powers:
Method 1: Prime factorization
1,024 can be expressed as 2^10. This means we can write it as (2^5) * (2^5), since (2^5) * (2^5) = 2^10 = 1,024.
Method 2: Dividing by powers of 2
We can start with 1,024 and repeatedly divide it by powers of 2 until we reach 1.
1,024 divided by 2 is 512, which can be expressed as 2^9.
512 divided by 2 is 256, which can be expressed as 2^8.
256 divided by 2 is 128, which can be expressed as 2^7.
128 divided by 2 is 64, which can be expressed as 2^6.
64 divided by 2 is 32, which can be expressed as 2^5.
32 divided by 2 is 16, which can be expressed as 2^4.
16 divided by 2 is 8, which can be expressed as 2^3.
8 divided by 2 is 4, which can be expressed as 2^2.
4 divided by 2 is 2, which can be expressed as 2^1.
2 divided by 2 is 1, which can be expressed as 2^0.
Therefore, 1,024 = 2^10 = (2^0) * (2^1) * (2^2) * (2^3) * (2^4) * (2^5) * (2^6) * (2^7) * (2^8) * (2^9).
Method 3: Using logarithms
We can use the power rule of logarithms to express 1,024 as the sum of logarithmic powers.
log base 2 of 1,024 = 10.
Therefore, log base 2 of 1,024 = log base 2 of (2^10).
To demonstrate that all three expressions are equivalent to 1,024, we can calculate the value of each expression:
Method 1: (2^5) * (2^5) = 2^(5+5) = 2^10 = 1,024.
Method 2: (2^0) * (2^1) * (2^2) * (2^3) * (2^4) * (2^5) * (2^6) * (2^7) * (2^8) * (2^9) = 1 * 2 * 4 * 8 * 16 * 32 * 64 * 128 * 256 * 512 = 1,024.
Method 3: log base 2 of (2^10) = 10.
Therefore, all three expressions are equivalent to 1,024.