Final answer:
To simplify the expression ((x^3y^(1/7))^(6/5))/(x^(4/5)y^(8/35)), we used the power of a power rule to expand the exponents and then subtracted like bases to get the simplified expression x^(14/5)y^(-2/35).
Step-by-step explanation:
The student is asked to simplify the expression ((x^3y^(1/7))^((6/5)))/(x^((4/5))y^((8/35))). When simplifying expressions involving exponents and division, we subtract exponents of like bases and correctly apply the power of a power rule. Let's walk through the problem step by step:
- Apply the power of a power rule: (x^3y^(1/7))^(6/5) = x^(3*(6/5))y^((1/7)*(6/5)) = x^(18/5)y^(6/35).
- Now divide the exponentials: x^(18/5) / x^(4/5) and y^(6/35) / y^(8/35).
- Subtract the exponents of like bases: x^(18/5 - 4/5) becomes x^(14/5) and y^(6/35 - 8/35) becomes y^(-2/35).
The simplified expression is x^(14/5)y^(-2/35).
We eliminate terms wherever possible and check that the answer is reasonable. The rule xPxQ = x^(P+Q) is applied to add exponents when multiplying with the same base, and to subtract exponents when dividing with the same base, which is utilized to simplify the given expression.