If a b c positive integers what is the positive square root of (a^2 b^16)(a c^2)^(-2)?

Rationale

a b^8 c^(-2)

To simplify the expression \((a^2 b^{16})(a c^2)^{-2}\), we need to apply the laws of exponents, which will help us find the positive square root correctly.

A (b^4 c^2) YOUR ANSWER

This option ignores the presence of \(a\) and the exponent for \(b\). The calculations show that the correct exponent for \(b\) after simplification should be \(8\), not \(4\). Hence, it fails to represent the correct form of the expression.

B (b^8 c^(-2)) YOUR ANSWER

While this option correctly identifies the exponent for \(b\), it neglects the factor \(a\) entirely. The expression derived from the original does not support a solution that lacks \(a\), making this choice incomplete.

C (a b^4 c^2) YOUR ANSWER

This choice introduces \(a\) but reduces the exponent of \(b\) incorrectly to \(4\). The correct exponent for \(b\) after simplification is \(8\), and it also does not properly account for the exponent on \(c\), which should be \(-2\).

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

This choice correctly incorporates all components of the expression, with \(a\) present and the exponents for \(b\) and \(c\) accurately represented as \(8\) and \(-2\), respectively. Thus, this is the correct simplification of the expression.

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

This option correctly includes \(a\) and \(b\) but miscalculates the exponent for \(c\), stating it as \(-3\) instead of \(-2\). This miscalculation disqualifies it as a valid answer.

Conclusion

The expression \((a^2 b^{16})(a c^2)^{-2}\) simplifies to \(a b^8 c^{-2}\). The correct solution must maintain all variables in their appropriate forms; thus, option D accurately reflects the required positive square root. Other choices either miscalculate or omit critical components, confirming D as the only valid answer.

If a b c positive integers what is the positive square root of (a^2 b^16)(a c^2)^(-2)?

a b^8 c^(-2)

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