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So i have an upcoming exam, and since no calculators is allowed, i was wondering if there is an approach to calculating the value of 2 to the power of some value?

For example,

2^4,2^5, 2^12, 2^13.

I know a way is as such where:

2^4 = 2x2x2x2 = 16
2^5 = 2x2x2x2x2 = 32
2^12 = 2x2x2x2x2x2x2x2x2x2x2x2 = 4096

But it would definitely take a long time for power of 12 and 13, so is there any faster approach on solving these without calculator?

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  • $\begingroup$ It's easier than you think. Just memorise squares and cubes of all natural numbers till 30 . ( Atleast!) There's no shortcut! If you memorise these values , then you should be able to answer faster than a calculator. For competitive exams , there are no shortcuts, only hardwork. All the best! Your calculation speed depends only on your practice. $\endgroup$ – Subhajit Halder Nov 10 '18 at 4:07
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You know $2^3 =8$ so $2^6 = 8^2=64$ and $2^8=16^2=256$, and $2^{10} = 4\times 256=1024$, $2^{13}=8192$,....

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Memorization and repeated squaring will get you there. If you want $2^{12}$, you can write it as $((2^3)^2)^2$ and then $2^3=8, 8^2=64, 64^2=4096$ Presumably you know the first two, so you only need to square $64$. If you want $2^{19}$ you can write it as $2^{16} \cdot 2^3=65536\cdot 8$. If you have a test like this (which I abhor) you should know the powers of $2$ up to at least $2^{16}$

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You can get to to $2^{10}=1024$ by doubling and squaring and you can get $2^{20}$ by knowing $2^{10} =1000 + 25 -1 = 100 + \frac {100}4 - 1$ and distributing.

So for example: $2^{17} = 2^{10}2^7 = (1000 + \frac {100}4 - 1)*2^7= 1000*2^7 + 100*2^5 - 2^7$.

$2^3= 8$ so $2^7 =8*8*2 = 64*2=128$. And $2^5 = 2^4*2 = 16*2=32$.

So $2^{17} = 128,000 + 3,200 - 128 = 131,100 - 28 = 131,070 + 2 = 131,072$

We can take it further $2^{20} = (1000 + 24)^2 = 1,000,000 + 48,000 + (25-1)^2 = 1,048,000 + 625 -50 + 1 = 1,048,576$. This gets iffy.

$2^{20} = 1,000,000 + \frac{100,000}2 - \frac{10,000}4 + 1,000 + \frac {100}2 + \frac {100}4 + 1$ so $2^{26}$ say is $64,000,000 + 3,200,000 - 160,000 + 64,000 + 3,200 + 1,600 + 64 = 67,108,864$.

It's not easy but it becomes "familiar looking"

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