Which of the following is equal to a1/2b1/3a1/4b-2/3, where a > 0 and b > 0 ?

Rationale

a^{3/4}b^{1/3}

To simplify the expression a^{1/2}b^{1/3}a^{1/4}b^{-2/3}, we first combine the exponents for each base. For 'a', we have (1/2 + 1/4) = 3/4, and for 'b', we get (1/3 - 2/3) = -1/3. Thus, the correct expression simplifies to a^{3/4}b^{-1/3}, which is equivalent to a^{3/4}/b^{1/3}.

A (a^3/4/b^1/3) CORRECT

This choice is correct, as explained above. The simplified expression a^{3/4}b^{-1/3} can indeed be rewritten as a^{3/4}/b^{1/3}.

B (a^1/8/b^2/9) YOUR ANSWER

This option is incorrect because the exponents for 'a' and 'b' do not match the derived values. The exponent of 'a' should be 3/4, and the exponent for 'b' should be -1/3, not 1/8 and 2/9, respectively.

C (-a^3/4b^1/3) YOUR ANSWER

This choice is incorrect as the expression obtained does not include a negative sign. The simplified expression, a^{3/4}/b^{1/3}, is strictly positive since a > 0 and b > 0.

D (-a^3/4b) YOUR ANSWER

The choice is incorrect because it does not follow exponent rules, changing a^(3/4) into a^(3/4) and b^(−1/3) into b, which is not equivalent to the original expression. It also incorrectly introduces a negative sign even though the expression is positive since a>0 and b>0.

Conclusion

The correct simplification of a^{1/2}b^{1/3}a^{1/4}b^{-2/3} leads to a^{3/4}/b^{1/3}, validating option A) as the correct answer. The other options either misrepresent the exponents or do not constitute valid expressions, emphasizing the importance of accurate manipulation of exponents in algebraic expressions.

Which of the following is equal to a1/2b1/3a1/4b-2/3, where a > 0 and b > 0 ?

a^{3/4}b^{1/3}

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