How to convert $10^n$ to scientific notation where $n \in \Bbb Q$?
For simplicity take the example $10^{7.44}$.
I know how to convert scientific notations to the power of $10$.
Like If we want to convert $3 \times 10^{-5}$ in power of $10$.
We can take $\log _{10}$ both sides of the equation $10^{n}=3 \times 10^{-5}$ and get the results .
So can we also do like this in reverse order to get the answer of my question?
Best Answer
Note that $10^{7.44}=10^{0.44} \times 10^7$, which is in scientific notation because $1\le10^{0.44}<10$ and $10^7$ is an integer power of $10$. However, in this context it would be more common to write $10^{0.44}$ as a decimal, and so we would write $$ 10^{7.44}=2.75 \times10^7 $$ where $10^{0.44}$ has been given to $3$ significant figures. I don't think there is a feasible way of computing $10^{0.44}$ without a calculator, although you can reason that $$ 2=8^{0.333}<10^{0.44}<16^{0.5}=4 $$ which gives you a very rough approximation.