Computing 2 to the power of some value without calculator

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?

• 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. – Subhajit Halder Nov 10 '18 at 4:07

You know $$2^3 =8$$ so $$2^6 = 8^2=64$$ and $$2^8=16^2=256$$, and $$2^{10} = 4\times 256=1024$$, $$2^{13}=8192$$,....

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 $$2^{16}$$

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"