2. Which of the following is equal to a1/2b1/3a1/4b-2/3, where a > 0 and b > 0 ?
Answer: A
a^3/4/b^1/3
The expression a1/2b1/3a1/4b-2/3 can be simplified to a^(1/2 + 1/4) / b^(2/3 - 1/3), which leads to a^(3/4) / b^(1/3).
A) a^3/4/b^1/3
This option correctly represents the simplified form of the given expression. By combining the exponents of 'a' and 'b' as per the rules of exponents, we find that a^(1/2 + 1/4) simplifies to a^(3/4) and b^(1/3) remains as is, making this option accurate.
B) a^1/8/b^2/9
This option is incorrect as it misrepresents the exponents of 'a' and 'b'. The exponents do not add up to 1/8 and 2/9, indicating a misunderstanding of exponent rules in the simplification process.
C) -a^3/4b^1/3
This choice is incorrect due to the negative sign in front of the expression. The original expression does not contain any negative exponents or terms that would introduce a negative value, thus making this option invalid.
D) -a^3/4b
Similar to option C, this choice is also incorrect because of the negative sign. Additionally, the exponent of 'b' is misrepresented, as the correct exponent should be 1/3, not 1.
Conclusion
Option A is definitively correct as it accurately reflects the simplification of the original expression using the laws of exponents. The other options fail either due to incorrect exponent calculations or the introduction of unnecessary negative signs, demonstrating a misunderstanding of the algebra involved.