How would I arrange these acids in order of strength? #"HClO"_2#, #"HIO"_4#, #"HClO"_3#, #"HClO"_4#, #"HBr"#, #"HCl"#, #"HI"#, #"NH"_4^+#

Well, you would have to be given a list of #K_a# values... Otherwise, you would memorize a list of 7 strong acids.

Just as a qualitative guess...

#ul("Acid"" "" "" "" "K_a" "" ")#
#"HI"color(white)(................)"Huge"#
#"HBr"color(white)(.........,..)"Large"#
#"HCl"color(white)(......,.....)"Pretty big"#
#"HClO"_4color(white)(..........)"Kinda big"#
#"HClO"_3color(white)(..........)"Kinda big"#
#"HIO"_4color(white)(...........)"Medium"#
#"HClO"_2color(white)(..........)"Medium"#
#"NH"_4^(+)color(white)(......,....)"Way small"#

Or with actual values...

#ul("Acid"" "" "" "" "K_a" "" "" "" "" "" "pK_a" ")#
#"HI"color(white)(................)10^10color(white)(.......................)-10#
#"HBr"color(white)(.........,..)10^9color(white)(.......................)-9#
#"HCl"color(white)(......,.....)10^7color(white)(.......................)-7#
#"HClO"_4color(white)(..........)39.8color(white)(......................)-1.6#
#"HClO"_3color(white)(..........)10color(white)(.......................)-1#
#"HIO"_4color(white)(...........)2.29 xx 10^(-2)color(white)(.............)1.64#
#"HClO"_2color(white)(...,....)1.1 xx 10^(-2)color(white)(................)1.96#
#"NH"_4^(+)color(white)(............)5.56 xx 10^(-10)color(white)(.............)9.26#

There is no simple rule to predict the strength of an acid. It is ALWAYS relative to whatever it is dissolved in, and relative to other acids.

In water, #K_a("H"_3"O"^(+)) = 1#, so if #K_a > 1#, the acid is strong in water.

Otherwise, you can compare acids.

OXO ACIDS

The more oxygens on the acid, the more they polarize the proton and the stronger the acid.

Hence, #K_a("HClO"_4) > K_a("HClO"_3) > K_a("HClO"_2) > K_a("HClO")#.

The less electronegative the central non-oxygen atom, the less it polarizes the proton, and the weaker the acid.

Hence, #K_a("HClO"_4) > K_a("HBrO"_4) > K_a("HIO"_4)#.

BINARY ACIDS

The larger the anion, the weaker the bond it forms with the proton and the stronger the acid.

Hence, #K_a("HI") > K_a("HBr") > K_a("HCl")# #">>"# #K_a("HF")#.

Besides those, #"NH"_4^(+)# is the conjugate acid of a weak base, #"NH"_3#, so of course it will be a weak acid.

[The conjugate acid of a weaker base is stronger of a weak acid than how strong of a base the weak base is. Therefore, the stronger the weak base, the weaker the conjugate weak acid.]