Middle school teacher here, and this is driving me crazy! Thanks!
In: 2
Because every power of 2 is double the one before, and adding two numbers together is the same as doubling them.
This is because another way to write this is just 2 * 2^9. Just like 4 + 4 is 2*4. 2^9 is just 2 times itself 9 times. That means 2^10 is 2 times itself 10 times or 2 * 2^9 or 2^9 + 2^9.
Because its 2^n when going to the next n you just double the previous. So 2^10 is double 2^9. Same would be true for any one like that. 2^5 + 2^5 = 2^6
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)
The difference between 2^9 and 2^10 is that 2^9 needs to be doubled one more time. Hence 2^9 + 2^9 = 2^10
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.
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
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.
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