Question

1. use factoring to determine whether each number is a perfect square, a perfect cube, or neither.

a) 225
b) 729
c) 1944
d) 1444
e) 4096
f) 13 824

Explanation:

Step1: Factorize 225

$225 = 3^2 \times 5^2 = (3 \times 5)^2 = 15^2$

Step2: Factorize 729

$729 = 3^6 = (3^2)^3 = 9^3 = (3^3)^2 = 27^2$

Step3: Factorize 1944

$1944 = 2^3 \times 3^5 = 2^3 \times 3^3 \times 3^2$

Step4: Factorize 1444

$1444 = 2^2 \times 19^2 = (2 \times 19)^2 = 38^2$

Step5: Factorize 4096

$4096 = 2^{12} = (2^6)^2 = 64^2 = (2^4)^3 = 16^3$

Step6: Factorize 13824

$13824 = 2^9 \times 3^3 = (2^3 \times 3)^3 = 24^3 = (2^4 \times 3^{1.5})^2$ (exponent of 3 is odd, so not a perfect square)

Answer:

a) Perfect square
b) Perfect square and perfect cube
c) Neither
d) Perfect square
e) Perfect square and perfect cube
f) Perfect cube