Methods to calculate powers in my head

Question

How can i calculate powers in my head, like small powers.

for example $0.5^3$, how can i work this out quickly and easily? or $4^5$

Answer 1

Not sure exactly of the scope of your question, but both of the powers you mentioned are powers of $2$.

$0.5^3 = \frac{1}{2^3} = \frac{1}{8} = 0.125$ (you can either memorise the decimal conversion or "recognise" it, see my note below).

and

$4^5 = (2^2)^5 = 2^{10} = 1024$.

These are very easy to work out once you've memorised the powers of $2$ to a reasonable exponent (I know them till $2^{20}$ without having to think at all, and I can work out a few more with a bit of thought).

You can then do "smaller" exponents of $4,8,16...$ just using the laws of exponents.

If you do a lot of computer work, the powers of $2$ will come about many times (e.g. in memory addressing), and you will start to remember them "naturally".

Powers of other bases can also be memorised, but they're not as easy, and frankly, I find them far less useful. Except maybe powers of $5$, which are also worth memorising up to at least $5^5$. The reason why memorising powers of $5$ is also useful is that it allows you to instantly "see" that the $\frac 18$ mentioned above is in fact $0.125$, which is basically $\frac{5^3}{10^3}$. So the reciprocals of the powers of $2$ are related to the powers of $5$. It's no accident because $2\times 5 = 10$, the decimal base.

Answer 2

Here are some tricks for computing $x^2$ :

• write $x=10a+b$ and use $(A+B)^2=A^2+2AB+B^2$ ; for instance $41^2=40^2+2\times 40\times 1+1=1681$.
• write $x=10a-b$; for instance $28^2=30^2-2\times 2\times 30+4=784$
• if $x=10a+5$, multiply $a$ by $a+1$ and write 25 right of it : $75^2=5625$ : discard the 5 it remains 7, 7x8=56 ; write 25 beside (this comes from ($10a+5)^2=100a^2+100a+25=100a(a+1)+25$)