Middle school teacher here, and this is driving me crazy! Thanks!
In: 2
This is a weird property of powers of 2, not a general property of exponents!
Think about what an exponent is. If we have `x^y`, that means we have `x*x` for as many times as `y` implies. So `2^y` is 2 times itself that many times.
2^1 = 2
2^2 = 2 * 2 = 2 + 2 = 4
2^3 = 4 * 2 = 4 + 4 = (2 + 2) + (2 + 2) = 2^2 + 2^2 = 8
2^4 = 2^3 * 2 = 2^3 + 2^3 = 8 + 8 = 16
It doesn’t work for other numbers.
3^1 = 3
3^2 = 3 * 3 = 9
But 3 + 3 = 6 != 9. Oops!
3^3 = 3 * 3 * 3 = 27
But 3^2 + 3^2 = 9 + 9 = 18, oops!
Basically it’s a fun trick because 2 * 2 and 2^2 are mathematically the same, which is not true for any other number.
Thanks all! I was in a meeting with some math and science teachers and we were trying to find a way to explain this to middle schoolers. Perfect explanations!
Because adding 1 to the exponent is the same as multiplying by the base, which in this case is 2.
This won’t work for other bases. E.g. 3^9 + 3^9 is not 3^10.
The left side of the equation has two terms, joined by a plus sign.
The number of terms just happens to be the exponential base of each term as well, `2`.
So the final addition, of `two` instances of the exponential term, has the equivalent effect of multiplying that exponential term by `2`.
In other words:
2^9 + 2^9 = 2 * ( 2^9 ) = 2^1 * 2^9 = 2^10
You’d find the same property in these equations:
3^9 + 3^9 + 3^9 = 3^10 (three terms)
4^9 + 4^9 + 4^9 + 4^9 = 4^10 (four terms)
… etc
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