Question 643932
I know that {{{2^10=1024}}}, and that helps me figure out that {{{highlight(2^9=512)}}} is the larges power of 2 with an integer exponent that is less than 665.
If/when I did not remember that,
I would calculate
{{{2^2=2*2=4}}}
{{{2^3=2*2*2=2^2*2=4*2=8}}}
{{{2^4=2*2*2*2=2^3*2=8*2=16}}}
and so on, so
{{{2^5=16*2=32}}}
{{{2^6=32*2=64}}}
{{{2^7=64*2=128}}}
{{{2^8=128*2=256}}} and
{{{2^9=256*2=512}}}
Of course I would not write all that just to calculate {{{2^9}}}. I would just write the results one after the other or one above the other:
2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,
and then I would count to find that 512 is the 9th number in that geometric sequence/progression,
so it is thwe product of {{{2*2*2*2*2*2*2*2*2}}} with nine twos, so it's {{{2^9}}}.