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I don't know what 55² is, and I wouldn't bother to compute it by hand.
But I do know:
a²-b² = (a-b)(a+b)
a² = (a-b)(a+b)+b²
Setting a=55 and b=5 we get
55² = (55-5)(55+5) + 5²
Suddenly it's obvious, not because of the numbers, but because of the structure.
The facility with numbers is an epiphenomenon.
2/n, n=2
@ColinTheMathmo
I didn't know about that formula, where does b comes from?
If I want to calculate another squared number how do I get b?
@Andres So the starting point is the "Difference of Two Squares" formula. So we have:
a²-b² = (a-b)(a+b)
You can go ahead and check that to make sure it's valid.
Then we can switch is around to get:
a² = (a-b)(a+b) + b²
Now when we want to square a number (call it "a") we have a free choice of "b" to see if we can make it easy.
Take 62² for example. Then choosing b=2 gives us:
62² = (62-2)(62+2) + 2²
That's (60 times 64) plus 4.
Now, 60 times 64 is 60 times 60 (which is 3600) plus 4 times 60 (which is 240) so we get:
62² = (60 times 64) plus 4
62² = (3600 + 240) + 4
62² = 3844 ... and we're done.
But ...
1/n