1

Problem

Simplify the expression \$((2x^3y^2)(-3xy^4)^2) / (6x^4y^3)^2\$.

2

Step

Given expression is

\$((2x^3y^2)(-3xy^4)^2) / (6x^4y^3)^2\$

3

Step

First, let’s simplify each part individually:

\$(2x^3y^2) = 2x^3y^2\$

\$(- 3xy^4)^2 = (-3)^2 \times (x)^2 \times (y^4)^2 = 9x^2y^8\$

\$(6x^4y^3)^2 = (6)^2 \times (x^4)^2 \times (y^3)^2 = 36x^8y^6\$

4

Step

Now plug these values back into the expression:

\$((2x^3y^2)(9x^2y^8)) / (36x^8y^6)\$

5

Step

Combine like terms by multiplying coefficients and adding exponents of variables:

\$(2 \times 9 \times x^3 \times x^2 \times y^2 \times y^8) / (36 \times x^8 \times y^6)\$

\$(18x^5y^(10)) / (36x^8y^6)\$

\$(x^5y^(10)) / (2x^8y^6)\$

6

Step

After simplification:

\$(1/2) \times (x^5 / x^8) \times (y^10 / y^6)\$

\$(1/2) \times x^(5 - 8) \times y^(10 - 6)\$

\$(1/2) \times x^(- 3) \times (y^4)\$

\$(1/2) \times y^4 / (x^3)\$

7

Solution

Therefore, the simplified expression is \$(1/2) \times y^4 / (x^3)\$.

8

Sumup

Please summarize steps

Choices

9

Choice-A

Option A inaccurately presents \$−(y^4)/(2x^3)\$, due to the inclusion of a negative sign, rendering it incorrect

Wrong \$ - y^4 / (2x^3)\$

10

Choice-B

It simplifies to \$y^3/(2x^2)\$, differing from our derived simplified form, thus incorrect

Wrong \$ y^3 / (2x^2)\$

11

Choice-C

Correct: It precisely performs the calculations with accuracy

Correct \$ (1/2) \times y^4 / (x^3)\$

12

Choice-D

\$-(y^3)/(2x^2)\$, is invalid as it diverges from the simplified expression due to calculation errors

Wrong \$ - y^3 / (2x^2)\$

13

Answer

Option

C

14

Sumup

Please summarize choices