Question 1

The positive square root of (a^2 b^16)(a c^2)^(-2) is a b^8 c^(-2).

To find the positive square root of the expression (a^2 b^16)(a c^2)^(-2), we first simplify it to a^2 b^16 / (a c^2)^2. The square root of this expression yields a b^8 c^(-2).

A) b^4 c^2

This option is incorrect as it does not account for the variable 'a' present in the expression. The simplification process shows that 'a' is a critical component that must be included in the final result.

B) b^8 c^(-2)

This option is also incorrect because it omits the variable 'a' entirely. While the powers of 'b' and 'c' are correctly represented, the absence of 'a' means it does not accurately reflect the expression we are simplifying.

C) a b^4 c^2

This option is incorrect as it misrepresents the powers of 'b' and 'c'. The simplification yields b^8 and c^(-2), so while 'a' is included, the powers for 'b' and 'c' do not match the correct calculation.

D) a b^8 c^(-2)

This option is correct as it accurately reflects the simplified expression. The square root of (a^2 b^16)(a c^2)^(-2) gives us a b^8 c^(-2), which includes the correct powers of each variable.

E) a b^8 c^(-3)

This option is incorrect because it misstates the power of 'c'. From our simplification, 'c' should have a power of -2, not -3, making this option invalid.

Conclusion

The correct answer, a b^8 c^(-2), comprehensively includes all necessary variables and correctly represents their respective powers derived from the simplification of the expression. Other options either omit critical components or miscalculate the powers, leading to incorrect conclusions. Thus, option D is the only one that accurately reflects the requirements of the question.