Final answer:
To find the value of n that makes the equation (2x⁹y) (4x²y¹⁰) - 8x!!y²⁰ true, we multiply the terms within the parenthesis and simplify. The resulting equation is 8x¹¹y¹¹. Assuming 'x!!' is a typo for 'x⁴', the answer is n = 11, which is not listed in the options. Thus, the correct option isn't provided in the question.
Step-by-step explanation:
The student is asking for the value of n that makes the given equation true: (2x⁹y) (4x²y¹⁰) - 8x!!y²⁰. To solve this, we need to multiply and simplify the terms. When we multiply the coefficients (2 and 4) and the variables with the same base, we add their exponents:
- Multiply the coefficients: 2 * 4 = 8
- Add the exponents of x: 9 + 2 = 11
- Add the exponents of y: 1 + 10 = 11
This results in 8x¹¹y¹¹. Now, to find the value of n for which the remaining part of the equation 8x!!y²⁰ is equivalent to 8x¹¹y¹¹, we focus on the x part. The notation x!! indicates that we're looking for a number n such that x to the n factorial (x⁴) equals x¹¹. This is only possible if n is equal to 11 since 11 factorial (11!) is the highest factorial less than or equal to 11 without being a decimal. Similarly, for y²⁰ to be equivalent to y¹¹, n must be 11. Thus, the correct answer is (d) 30, if we interpret 'x!!' as a typo and it should be 'x⁴'.