Final answer:
To simplify the expression a¹/³/(a¹/´)a⁻¹/², we subtract the exponents. After finding a common denominator and combining the terms, the simplified expression is a⁷/¹², with all variables assumed positive real numbers.
Step-by-step explanation:
The student is asking to simplify the expression a¹/³ / (a¹/´a⁻¹/²). When simplifying expressions with exponents, we use the rules of exponents to combine them correctly. The division of two exponential expressions with the same base allows us to subtract the exponents. Negative exponents signify inversion or placing the term in the denominator. Here's how to simplify the expression step by step:
- First, set up the division as a subtraction of exponents because they have the same base a.
- Since we have a¹/³ divided by a¹/´ multiplied by a⁻¹/², we subtract the exponents, keeping in mind that subtracting a negative is like adding a positive.
- Combine the exponents: 1/3 - (1/4 + (-1/2)). This requires finding a common denominator for the fractions.
- The common denominator for 3, 4, and 2 is 12, so rewrite the exponents with this common denominator: 4/12 - (3/12 - 6/12).
- Perform the subtraction: 4/12 - 3/12 + 6/12 = 7/12.
- Therefore, the simplified expression is a⁷/¹².
Remember, we leave the exponent positive to follow the instruction of using only positive exponents, and we assume all variables are positive real numbers as per the instructions.