On 10/07/2020 18:43, Henry Baker wrote:
True.
Now do it like a high schooler...
Step 1: either use a calculator or do some rough pencil-and-paper approximation to guess that the answer might be 2. Step 2: play around with those two square-rooted quantities a bit, finding that their product is the cube root of 8, namely 2 [citation needed]. Step 3: if we call the one with the + sign "A" then this means we're trying to prove that A - 2/A = 2. Step 4: Wait, we know what the solutions of that are; they're [pulls out quadratic formula] 1 +- sqrt(3). So that ought to mean that A is 1 + sqrt(3). Step 5: Waaait, we can check that easily. What's the _cube_ of 1 + sqrt(3)? [scribbles] 10 + 6 sqrt(3), which is indeed the same thing as sqrt(108) + 10. Step 6: OK, so we've just proved that our first quantity is 1 + sqrt(3). So is the second one sqrt(3) - 1? That'll be true if _that_ cubes up to sqrt(108)-10. [scribbles] Yes, it does. (For extra points, "Of course it does" without the scribbling, but that might be a bit much to expect of a high-schooler.) And we're done. Nothing there should be beyond the wit of, say, a bright 15-year-old. -- g