How to calculate $10^{0.4}$ without using calculator How to calculate $10^{0.4}$  without using calculator or if not what is the closest answer you can get just using pen and paper within say $2$ min?
 A: When doing rough logarithms or antilogarithms under conditions like
the ones in the question, it helps to remember a few frequently-used
approximations:
\begin{align}
10^{0.301} & \approx 2 \\
10^{0.5}   & \approx 3.16
\end{align}
It may be easier to remember the first fact if you keep in mind that
$2^{10} = 1024$ is a little greater than $10^3$.
Then observe that 
$$10^{0.4} = \frac{10}{\left(10^{0.3}\right)^2}
 \approx \frac{10}{2^2} = 2.5. \tag1$$
If you want to refine this, note that the rough approximation
$10^{0.3} \approx 2$ has an error between $2\%$ and $2.5\%$,
so Equation $(1)$ has divided by too much and the result should
be about $4\%$ or $5\%$ greater than shown.
A: You are looking for a quick method that requires the minimum of arithmetic.
In the absence of fifth powers of integers which are close to 100, we look for two integers the ratio of the fifth powers of which are close to 100, and we are fortunate that $$\frac{5^5}{2^5}=\frac{3125}{32}=\frac{3200}{32}-\frac{75}{32}$$
We can therefore consider a simple linear approximation of the form $$f(x+h)\simeq f(x)+hf'(x)$$ using the function $$f(x)=x^{\frac 15}\Rightarrow f'(x)=\frac 15 x^{-\frac 45}$$
Therefore $$100^{\frac 15}=f(100)=f\left(\frac{3125}{32}+\frac{75}{32}\right)\simeq 2.5+\frac {75}{32}\times\frac 15\times\left(\frac{3125}{32}\right)^{-\frac 45}$$
$$\simeq 2.5+\frac{15}{32}\times\frac{16}{625}$$
$$\simeq 2.5+\frac{3}{250}$$
$$\simeq 2.512$$
This is correct to 3 decimal places and easily doable in two minutes.
