Question

Which are differences of perfect cubes? Check all that apply.

Answer 1

we know that

A perfect cube is a whole number which is the cube of another whole number

so

A number is a perfect cube, if the cubic root of the number is equal to a whole number

case 1)

$9a^{3} - 27$

$9a^{3} - 27 = 9a^{3}-3 ^{3}$

in this problem the cubic root of $9$ is not a whole number

therefore

The case 1) is not a differences of perfect cubes

case 2)

$125a^{12} - 72$

$125a^{12} - 72=5^{3}(a^{4})^{3} -\sqrt[3]{72}$

in this problem the cubic root of $72$ is not a whole number

therefore

The case 2) is not a differences of perfect cubes

case 3)

$216a^{6}- 27y^{3}$

$216a^{6}- 27y^{3} = 6^{3}(a^{2})^{3}- 3^{3}y^{3} = (6a^{2})^{3} - (3y)^{3}$

therefore

The case 3) is a differences of perfect cubes

case 4)

$8a^{15} - 27$

$8a^{15} - 27 = 2^{3}( a^{5})^{3} - 3^{3} = (2a^{5} )^{3} - 3^{3}$

therefore

The case 4) is a differences of perfect cubes

case 5)

$225a^{3} - 216$

$225a^{3} - 216 =5^{3}a^{3} -6^{3}= (5a)^{3} - 6^{3}$

therefore

The case 5) is a differences of perfect cubes

therefore

the answer is

$216a^{6}- 27y^{3}$

$8a^{15} - 27$

$225a^{3} - 216$

Answer 2

We are to determine if the given are differences of perfect cubes and in order to do so, we simply have to calculate for the cube root of the terms in the given.

(a)   9a³ - 27 = ((∛9)a)³ - 3³   ; NO
(b)   125a¹² - 72 = (5a⁴)³ - (∛72)³ ; NO
(c)   216a⁶ - 27y³ = (6a²)³ - (3y)³ ; YES
(d)   8a¹⁵ - 27 = (2a⁵)³ - (3)³ ; YES
(e)   225a³ - 216 = (15a)³ - (6)³  ; YES

Therefore, the answers are the third, fourth, and fifth choices.