In general, any number $x$ is equal to $10^{\log_{10} x}$. In particular, for your number:
$$1.625×2^{35} = 10^{\log_{10} \left( 1.625\cdot2^{35}\right)}.$$
You can use this to put your number in the form your want. Start by taking the base-10 logarithm of $1.625\cdot 2^{35}$:
$$\begin{align} \log_{10}\left( 1.625×2^{35} \right)& = \log_{10} 1.625 + 35\cdot \log_{10} 2 \\ &= 0.2108534 + 35\cdot 3.3219809 \\ &= 10.7468 \end{align} $$
And then the answer is:
$$\begin{align} 10^{10.7469032} & =10^{0.7469032} \cdot 10^{10} \\ & = 5.5834575\cdot 10^{10} \end{align} $$
The best way to calculate the logarithm and the inverse logarithm ($10^x$) is with a digital computer of some type, maybe a pocket calculator. For example, how do you find out that $\log_{10} 1.625 \approx 0.2108$? You put 1.625 into your calculator and push the log
button; you calculate $10^{0.7469032}$ similarly.