Is there a fast and easy way to calculate high powers mentally, e.g. $ 67^{81} $?

Question

I have done research into vedic mathematics and I was wondering if it's possible that a faster method exists than the one I already know which involves Pascal's triangle.

Answer 1

As pointed out in the comments, I'm assuming you don't actually need all the digits, just an approximation. For this you can use $x^y = 10^{y \log_{10} x }$, and then break down $x$ to it's nearest power of ten - this will work well when $x$ is large. In your example: $$67^{81}=10^{81 \log_{10} 67}$$ $$\log_{10} 67=\log_{10} (0.67\times100)=2+\log_{10} (0.67) =2+\frac{\ln 0.67}{\ln 10}$$ Using the Taylor series, you can calculate: $$\ln 0.67=\ln(1-0.33)\approx-0.33-\frac{0.33^2}{2}-\frac{0.33^3}{3}\approx-0.396$$ And now using $\ln 10\approx 2.3$, we have: $$81\log_{10} 67 \approx 148$$ Giving you $10^{148}$, which is really close to the correct $\approx8.2\times 10^{147}$.