Middle school teacher here, and this is driving me crazy! Thanks!
In: 2
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
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.
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!
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.
A base is multiplied to itself using the exponent as a reference for that number of times it should be multiplied. So, it’s just repeated multiplication. Multiplication can be defined as repeated summation. And therefore, you can show from the bases of repeated multiplication, simplified down to the equivalent repeated addition of them, that the base 2 on each side of the equation is merely the repeated addition of itself the same number of times.
2^9 is 2 multiplied to itself 9 times and is equivalent to adding 256 “2’s” together. Since there are two terms, then on the left side are a combined 512 “2’s” added together. Or, 2*512.
2^10 is 2 multiplied to itself 10 times and is equivalent to adding 512 “2’s” together. Or, 2*512.
2^9 = 2*2*2*2*2*2*2*2*2. For any number x, x+x = 2*x, so 2^9 + 2^9 = 2*2^9 = 2*2*2*2*2*2*2*2*2*2 = 2^10.
For a similar reason, 3^9 + 3^9 + 3^9 = 3^10.
You can rewrite 2^9 + 2^9 as 2*2^9 that is equal to 2^10 Why it is equal is because of the definition of the exponent 2^9 =2*2*2*2*2*2*2*2*2 by definition so if you multiply it by 2 you get 2^10
You can do the same for any base you just need to have add together the same number as the base
3^4 +3^4 + 3^4 = 4^5
You can do that with base 10 to so 10^2 + 10^2 + 10^2 + 10^2 + 10^2 + 10^2 + 10^2 + 10^2 + 10^2 + 10^2 =10^3 That is equalvnet to 10 * (10^2) = 10 * 100 =1000
Writing it out is probably easier…
2^2 = 2×2 = 4
2^3 = 2x2x2 = 8
2^4 = 2x2x2x2 = 16
……..
2^9 = 2x2x2x2x2x2x2x2x2 = 512
2^10 = 2x2x2x2x2x2x2x2x2x2 = 1024
2^9 + 2^9 is 2 times 2^(9).
2 is 2^(1).
And 2^(1) times 2^(9) is 2^(10) by the usual rules of exponents.
When you do 2^9 + 2^9 it’s the same as 2 * 2^9 (as multiplication is just repeated addition) which is the same as 2^10 (as powers are just repeated multiplication)
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