I need to know how to calculate 10^ a decimal power, like 10^-7.4, without a calculator, in as simple a way as possible, since I will be doing questions which only allow me about a minute to a minute and a half each. Does anyone know a good technique for this?
Edit: I'm not allowed to use any other resources besides a pencil/paper/my head.
Best Answer
Notice that $$10^{-7.4} = 10^{-7} \cdot 10^{-0.4}$$
Now take a logarithm of the second part:
$$\ln 10^{-.4} = -0.4 \ln 10 \approx -0.4 \cdot 2.3 = -0.92$$ after memorizing what the log of 10 is. Then exponentiate:
\begin{align*} 10^{-.4} = e^{-.92} &= \sum_{k = 0}^{\infty} \frac{(-.92)^k}{k!} \\ &\approx 1 - .92 + \frac{.92^2}{2} - \frac{.92^3}{6} + \frac{.92^4}{24} \end{align*}
correct to at least two decimals: This is an alternating series, and so the error can be estimated by the next term, which is substantially less than $1/120$. Calculating this by hand should only take little bit of time, and I get about $0.40$ (although I did use a calculator).
So my answer would be $4.0 \times 10^{-8}$, which compares well to the more precise answer of $3.98107 \times 10^{-8}$ from a calculator. Had I computed one more term in the series expansion, I would have gotten $3.978 \times 10^{-8}$, which is quite good.
Alternative solution: Memorize that $10^{0.1} \approx 1.259$ and go from there.