Base game

$100_2+100_8+100_{10}+100_{16}$

$=(\ 2^2\times 1\ )+(\ 8^2\times 1\ )+(\ 10^2\times 1\ )+(\ 16^2\times 1\ )$

$=\ 4_{10}+64_{10}+100_{10}+256_{10}\ =\ 424_{10}$

100(base 2) is 100 in binary form = 4

100(base 8) is 100 in octal form = 64

100(base 10) is 100 in decimal form = 100

100(base 16) is 100 in hexadecimal form = 256

4 + 64 + 100 + 256 = 424

For any base $x$ , $100_x = x^2_{10}$ .

Then $2^2+8^2+10^2+16^2=424$ .

100 2 ( x indicates base number) is in binary. So, decimal form of this number is : (1 * 2^2) + (0 * 2^1) + (0 * 2^0) = 4 + 0 +0 = 4

100_8 is in octal. So, decimal form of this number is: (1 * 8^2) + (0 * 8^1) + (0 * 8^0) = 64 + 0 + 0 = 64

100_10 is already in decimal. No conversion required here. So, the number is: = 100

100_16 is in hexadecimal. So, the decimal form of this number is: (1 * 16^2) + (0 * 16^1) + (0 * 16^0) = 256 + 0 + 0 = 256

Thus, the number is:

4 + 64 + 100 + 256 = 424

OR

1 2^2+1 8^2+1 10^2+1 16^2=424

AND

2²+8²+10²+16² = 4+64+100+256 = 424

$100_2+100_8+100_{10}+100_{16}=2^2+8^2+10^2+16^2=4+64+100+256=\boxed{\large{424}}$

2^2 + 8^2 + 10^2 + 16^2 = 4 + 64 + 100 + 256 = 424