Question

use the sum of cubes identity to find the factors of the expression $8x^6 + 27y^9$. write your answer in the form of the expression on the right of the sum of cubes identity. enter the correct answer in the box.

Explanation:

Step1: Identify sum of cubes form

Recall sum of cubes identity: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.
Rewrite $8x^6 + 27y^3$ as cubes:
$8x^6 = (2x^2)^3$, $27y^3 = (3y)^3$
So $a=2x^2$, $b=3y$

Step2: Apply the identity

Substitute $a$ and $b$ into the identity:
First factor: $a + b = 2x^2 + 3y$
Second factor: $a^2 - ab + b^2 = (2x^2)^2 - (2x^2)(3y) + (3y)^2$

Step3: Simplify the second factor

Calculate each term:
$(2x^2)^2 = 4x^4$, $(2x^2)(3y)=6x^2y$, $(3y)^2=9y^2$
So the second factor is $4x^4 - 6x^2y + 9y^2$

Answer:

$(2x^2 + 3y)(4x^4 - 6x^2y + 9y^2)$