Which of the following is equivalent to √(a ^ 6 * b ^ 12) - √(a ^ 3) where a and b are positive constants?

Rationale

√(a ^ 6 * b ^ 12) - √(a ^ 3) simplifies to a^3b⁹.

To simplify the expression, we first calculate √(a ^ 6 * b ^ 12), which becomes a^3b^6. Then, subtracting √(a ^ 3) or a^(3/2) results in a^3b^6 - a^(3/2). Since a^(3/2) = a^(1.5), we can rewrite the expression further to arrive at the simplest form of a^3b^9.

A (a^2^b4-a) YOUR ANSWER

This option incorrectly combines terms and does not match the simplified expression. The term a^2b^4 does not correctly reflect the powers derived from √(a ^ 6 * b ^ 12) and √(a ^ 3).

B (a^3b⁹-a) YOUR ANSWER

While this option includes a^3b^9, the subtraction of 'a' is incorrect. The expression must retain the correct terms without additional subtractions or modifications that do not arise from the simplification process.

C (a²b^4+-1) YOUR ANSWER

This option contains the incorrect terms and operations, introducing a constant and incorrectly modifying the powers of 'a' and 'b'. The simplification process does not yield a constant term, which disqualifies this choice.

D (a^3b⁹-1) YOUR ANSWER

Like option B, this answer includes the correct term a^3b^9 but mistakenly subtracts 1. The original expression does not yield a constant subtraction, making this choice incorrect.

E (a^3b⁹) CORRECT

This choice correctly represents the simplified form of the original expression after performing the necessary calculations and subtractions.

Conclusion

The correct simplification of √(a ^ 6 * b ^ 12) - √(a ^ 3) results in a^3b^9. All other choices either introduce incorrect terms or modify the necessary components of the expression, highlighting the importance of careful algebraic manipulation in reaching the correct answer.

Which of the following is equivalent to √(a ^ 6 * b ^ 12) - √(a ^ 3) where a and b are positive constants?

√(a ^ 6 * b ^ 12) - √(a ^ 3) simplifies to a^3b⁹.

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