Alright, let's simplify each expression step-by-step.
### Part a: Simplify [tex]\((14b - 4a) - 6(2a - \frac{1}{2}b + 2b)\)[/tex]
First, distribute the 6 within the parentheses:
[tex]\[
6(2a - \frac{1}{2}b + 2b) = 6 \cdot 2a - 6 \cdot \frac{1}{2}b + 6 \cdot 2b = 12a - 3b + 12b
\][/tex]
Combine like terms in the expression within the parentheses:
[tex]\[
12a - 3b + 12b = 12a + 9b
\][/tex]
Rewrite the original expression:
[tex]\[
(14b - 4a) - (12a + 9b)
\][/tex]
Distribute the negative sign:
[tex]\[
14b - 4a - 12a - 9b
\][/tex]
Combine like terms:
[tex]\[
(14b - 9b) + (-4a - 12a) = 5b - 16a
\][/tex]
So, the simplified expression is:
[tex]\[
5b - 16a
\][/tex]
### Part b: Simplify [tex]\(-\left(2x^2y^3 - 4x^3y^5 + x^2y^3\right) - x^2 y^3 \cdot 4xy^2\)[/tex]
First, simplify inside the parentheses:
[tex]\[
2x^2y^3 - 4x^3y^5 + x^2y^3 = 2x^2y^3 + x^2y^3 - 4x^3y^5 = 3x^2y^3 - 4x^3y^5
\][/tex]
Distribute the negative sign:
[tex]\[
-(3x^2y^3 - 4x^3y^5) = -3x^2y^3 + 4x^3y^5
\][/tex]
Next, simplify the product:
[tex]\[
x^2 y^3 \cdot 4xy^2 = 4x^3y^5
\][/tex]
Combine the expressions:
[tex]\[
-3x^2y^3 + 4x^3y^5 - 4x^3y^5
\][/tex]
Combine like terms:
[tex]\[
-3x^2y^3 + 0 \cdot x^3y^5 = -3x^2y^3
\][/tex]
So, the simplified expression is:
[tex]\[
-3x^2y^3
\][/tex]
### Part c: Simplify [tex]\(3(x - y) + 2(-2x + 6y)\)[/tex]
Distribute the constants inside the parentheses:
[tex]\[
3(x - y) = 3x - 3y
\][/tex]
[tex]\[
2(-2x + 6y) = -4x + 12y
\][/tex]
Combine the expressions:
[tex]\[
3x - 3y - 4x + 12y
\][/tex]
Combine like terms:
[tex]\[
(3x - 4x) + (-3y + 12y) = -x + 9y
\][/tex]
So, the simplified expression is:
[tex]\[
-x + 9y
\][/tex]
### Part d: Simplify [tex]\(2yx^3 \times 6xy^4 - \frac{16x^2y}{2x}\)[/tex]
First, simplify the product:
[tex]\[
2yx^3 \times 6xy^4 = 12yx^4y^4 = 12x^4 y^5
\][/tex]
Simplify the division:
[tex]\[
\frac{16x^2y}{2x} = \frac{16}{2}x^{2-1}y = 8xy
\][/tex]
Combine the expressions:
[tex]\[
12x^4y^5 - 8xy
\][/tex]
So, the simplified expression is:
[tex]\[
12x^4y^5 - 8xy
\][/tex]
