Question

which are differences of perfect cubes? Check all that apply. 9a3 – 27 125a12 – 72 216a6 – 27y3 8a15 – 27 225a3 – 216

Answer 1

ANSWER

The correct choices are:

${8a}^{15} - 27$

and

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

EXPLANATION

A difference of perfect cube is of the form:

${(m)}^{3} - {(n)}^{3}$

We can rewrite

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

Try rewriting the terms as powers to an exponent of 3.

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

This implies that:

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

Hence this is a difference of perfect cube.

Also, we can rewrite

${8a}^{15} - 27$

as

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

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