Multiply and simplify completely
1. (x-2y)^3
Simplify
2. 2(-3x^3 y^2)^3
45
To multiply and simplify the expression (x-2y)^3, we'll use the binomial theorem. According to the binomial theorem, (x-2y)^3 can be expanded as follows:
(x-2y)^3 = C(3,0) * (x)^3 * (-2y)^0 + C(3,1) * (x)^2 * (-2y)^1 + C(3,2) * (x)^1 * (-2y)^2 + C(3,3) * (x)^0 * (-2y)^3
where C(n,r) represents the binomial coefficient (n choose r), which is given by the formula C(n,r) = n! / (r! * (n-r)!).
Let's simplify each term step by step:
1. (x)^3 * (-2y)^0 = x^3 * 1 = x^3
2. C(3,1) * (x)^2 * (-2y)^1 = 3 * (x^2) * (-2y) = -6x^2y
3. C(3,2) * (x)^1 * (-2y)^2 = 3 * x * (4y^2) = 12xy^2
4. (x)^0 * (-2y)^3 = 1 * (-8y^3) = -8y^3
Combining all the simplified terms, we have:
(x-2y)^3 = x^3 - 6x^2y + 12xy^2 - 8y^3
Now, let's simplify the expression 2(-3x^3 y^2)^3:
To simplify this expression, we'll first simplify the exponent by applying the power rule:
(-3x^3 y^2)^3 = (-3)^3 * (x^3)^3 * (y^2)^3 = -27x^9y^6
Next, we'll distribute the coefficient 2 to the simplified expression:
2 * (-27x^9y^6) = -54x^9y^6
Therefore, the simplified expression is -54x^9y^6.