Answer:
After comparing both sides of the equation, we can see that they are not equal. Therefore, the given equation is not true.
If you have any further questions, please let me know.
Explanation:
To prove the given equation:
(2x + 3y)(2x - 3y)(4x^2 + 6xy + 9y^2)(4x^2 - 6xy + 9y^2) = 64x^6 - 729y^6
We can start by expanding both sides of the equation and simplifying:
Expanding the left side:
(2x + 3y)(2x - 3y) = 4x^2 - 9y^2
(4x^2 + 6xy + 9y^2)(4x^2 - 6xy + 9y^2) = (16x^4 - 54x^2y^2 + 81y^4)
Now, let's multiply the two expressions we obtained:
(4x^2 - 9y^2)(16x^4 - 54x^2y^2 + 81y^4)
Using the difference of squares formula (a^2 - b^2 = (a + b)(a - b)), we can rewrite the first term:
(4x^2 - 9y^2) = (2x)^2 - (3y)^2 = (2x + 3y)(2x - 3y)
Substituting this back into the equation:
(2x + 3y)(2x - 3y)(16x^4 - 54x^2y^2 + 81y^4)
Now, let's simplify the expression further:
(2x + 3y)(2x - 3y)(16x^4 - 54x^2y^2 + 81y^4)
= (2x + 3y)(2x - 3y)(4x^2)^2 - (3y)^2)
= (2x + 3y)(2x - 3y)(4x^2)^2 - (3y)^2)
= (2x + 3y)(2x - 3y)(16x^4 - 9y^2)
Expanding the remaining terms:
= (2x + 3y)(2x - 3y)(16x^4) - (2x + 3y)(2x - 3y)(9y^2)
= (2x + 3y)(2x - 3y)(16x^4) - (2x + 3y)(2x - 3y)(9y^2)
= (16x^4)(2x + 3y) - (9y^2)(2x + 3y)
= 32x^5 + 48x^4y - 18xy^2 - 27y^3
Now, let's simplify the right side of the equation:
64x^6 - 729y^6
The equation becomes:
32x^5 + 48x^4y - 18xy^2 - 27y^3 = 64x^6 - 729y^6
After comparing both sides of the equation, we can see that they are not equal. Therefore, the given equation is not true.
If you have any further questions, please let me know.