How do you use factoring to determine whether which of these numbers is a perfect square, a perfect cube, or neither: #225, 729, 1944, 1444, 4096 and 13824#?

Question

Hazel Anglin · Accepted Answer

Write each number as the product of its prime factors.

A square number has its prime factors in pairs so that the index is an even number ( a multiple of #2#)

A cube has its prime factors in groups of #3#, so the indices are multiples of #3#

#225= 3xx3xx5xx5 = 3^2 xx 3^2 = 15^2" "larr# a perfect square

#729 = 3xx3xx3xx3xx3xx3= 3^6 = 9^3" "larr# a perfect cube

#1944 = 2xx2xx2 xx3xx3xx3xx3xx3 = 2^3 xx3^5larr# neither

#1444 = 2xx2xx19xx19 = 2^2 xx 19^2 = 38^2larr# a perfect square

#4096 = 2xx2xx2xx2xx2xx2xx2xx2xx2xx2xx2xx2= 2^12#

It is a perfect square and a perfect cube.

#4096 = (2^6)^2 =64^2" "and" "4096 = (2^4)^3 = 16^3 #

#13,824 = 2^9 xx 3^3 = (2^3xx3)^3 = 24^3" "larr# a perfect cube