Final answer:
To find the given product, the square roots are expressed as fractional powers and the expression is simplified before squaring. Upon squaring, the exponents are multiplied and the result is 80x^4√30, which is option a).
Step-by-step explanation:
To find the product of the expression [4x√(5x^2)√6]^2, we first simplify the expression inside the brackets before raising it to the power of 2. The square roots can be expressed as fractional exponents, and we remember that when multiplying exponents with the same base, we add the exponents.
√5 can be expressed as 5^(1/2) and √6 as 6^(1/2), both of which are raised to the power of 2 when the entire expression inside the brackets is squared. Similarly, 5x^2 can be written as (5^(1/2)x^(2/2)). Now we combine the like bases and apply the power of a product rule ((ab)^2 = a^2b^2) when we square the entire expression:
First, simplify inside the brackets: 4x√(5x^2)√6 = 4x·5^(1/2)x·6^(1/2) = 4x^2·5^(1/2)·6^(1/2).
Then raise to the power of 2: [4x^2·5^(1/2)·6^(1/2)]^2 = 16x^4·5·6 = 80x^4√30.
Therefore, the correct answer is a) 80x^4√30.