1. Which of the following is equivalent to √(a ^ 6 * b ^ 12) - √(a ^ 3) where a and b are positive constants?
Answer: E
√(a ^ 6 * b ^ 12) - √(a ^ 3) simplifies to a^3b^9
By simplifying the expression, we find that √(a ^ 6 * b ^ 12) equals a^3b^6, and thus, the entire expression becomes a^3b^6 - a, which can be rewritten as a^3b^9.
A) a^2^b4-a
This option does not hold true because it incorrectly combines the variables and powers. The expression does not equate to the simplified form derived from √(a ^ 6 * b ^ 12) - √(a ^ 3).
B) a^3b⁹-a
While this option is close, it is not entirely accurate. The correct expression derived from the given equation does not include the additional subtraction of 'a', thus making this option incorrect.
C) a²b^4+-1
This option is incorrect as it misrepresents the values of 'a' and 'b'. The simplification clearly does not lead to a form that includes 'b^4' nor the subtraction of 1, making it an invalid choice.
D) a^3b⁹-1
This option also fails to represent the correct relationship derived from the original expression. The subtraction of 1 does not correspond to the simplification of the square roots, rendering this choice incorrect.
E) a^3b⁹
This is the correct answer because it accurately reflects the result of simplifying the expression √(a ^ 6 * b ^ 12) - √(a ^ 3). The simplification confirms that the expression resolves to a^3b^9.
Conclusion
The correct answer is E, as it directly corresponds to the simplified form of the original expression. All other options either misrepresent the powers of 'a' and 'b' or incorrectly introduce additional terms, leading to their invalidity in this context.