2260.11 – Huge Difference of Squares

Find among the suggested answer below the sum of the squares of all real numbers that satisfy the equation

x256−25632=0x^{256} - 256^{32} = 0

  1. 88

  2. 128128

  3. 512512

  4. 65,53665,536

  5. 2×256322 \times 256^{32}

Solution

The real problem is solving for xx. First, simplify the equation to

x256=25632. x^{256} = 256^{32}.

At this point you could calculate 25632256^{32} and take the 256th root to find xx. But it's easier to use 256=28256 = 2^8 to rewrite the equation:

x256=25632=(28)32=2256.x^{256} = 256^{32} = (2^8)^{32} = 2^{256}.

In other words, x=±2x = ±2. We must now square these two solutions and add:

22+(−2)2=4+4=8.2^2 + (-2)^2 = 4+4 = 8.

The solution is a.