Changing Between Bases: Special Case

A shortcut exists for converting between bases when one base is an integer power of the other (e.g., base-2 to base-8, since $8=2^3$).

To convert from a smaller base to a larger base ($m$ to $n=m^a$):

Group the digits of the number in base $m$ into sets of $a$. Convert each group from base $m$ to base 10, which gives you the corresponding digit in base $n$. For example, to convert $110001011_2$ to base-8, group into threes: $[110 | 001 | 011]$. Convert each group: $110_2=6$, $001_2=1$, $011_2=3$. The answer is $613_8$.

To convert from a larger base to a smaller base ($n$ to $m$):

Convert each digit of the base-$n$ number into an $a$-digit number in base $m$. For example, converting $643_9$ to base-3 ($9=3^2$, so $a=2$): $6_9 = 20_3$, $4_9 = 11_3$, $3_9 = 10_3$. The answer is $201110_3$.