Factor:
(x^3)(z^6)-27y^9
81x^4-16y^4
2x^3+16y^3
You have the usual sum/difference of squares/cubes.
(xz^2)^3 - (3y^3)^3
= (xz^2-3y^3)((xz^2)^2 + (xz^2)(3y^3) + (3y^3)^2)
Similarly, the others are
(9x^2)^2 - (4y^2)^2
and
2(x^3 + (2y)^3)
Just apply your formulas. If you get stuck, come on back and say where.
To factor these expressions, we need to find the common factors and use some algebraic techniques.
1. (x^3)(z^6) - 27y^9:
We can see that both terms have a power of 3, x^3 as the common factor. And in the second term, 27 can also be expressed as (3^3). Using the difference of cubes formula, we know that a^3 - b^3 = (a - b)(a^2 + ab + b^2). Applying this formula, we have:
(x^3)(z^6) - 3^3(y^3)^3 = (xz^2 - 3y^3)(x^2z^4 + 3xy^3z^2 + 9y^6).
2. 81x^4 - 16y^4:
This expression doesn't have any common factors except for 1. However, it is a special case known as the difference of squares, which can be further factored using the difference of squares formula: a^2 - b^2 = (a + b)(a - b). Applying this formula to our expression, we have:
81x^4 - 16y^4 = (9x^2 + 4y^2)(9x^2 - 4y^2).
3. 2x^3 + 16y^3:
At first glance, it seems that there are no common factors. However, we can factor out a 2 from both terms. So, we have:
2(x^3 + 8y^3).
Unfortunately, the expression x^3 + 8y^3 cannot be factored any further since it is in the form of a sum of cubes. Factoring a sum of cubes follows a different formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2).