Final answer:
To simplify (32x⁻³⁰y³⁵)²/⁵, we raise each term inside the parentheses to the power of 2 and then divide the exponents by 5. This results in a simplified expression of 1024y¹⁴/x¹² with all exponents being positive.
Step-by-step explanation:
To simplify the expression (32x⁻³⁰y³⁵)²/⁵ using only positive exponents, we must apply the rules of exponents to each part of the expression. Remember that when you raise a power to a power, you multiply the exponents. Additionally, when dividing powers with the same base, you subtract the exponents.
First step: Apply the exponent outside the parentheses to each term inside.
(32)^2 becomes 32², which is 1024.
x⁻³⁰² becomes x⁻³⁰⋅², which is x⁻¶⁰.
y³⁵² becomes y³⁵⋅², which is y⁷⁰.
Second step: Simplify the exponents by dividing by 5, as indicated by the ²/⁵. This results in:
1024ⁱ⁵
For x⁻¶⁰, we divide -60 by 5, giving us x⁻¹².
For y⁷⁰, we divide 70 by 5, giving us y¹⁴.
Final step: Rewrite the expression with positive exponents by bringing x⁻¹² to the denominator:
1024y¹⁴/x¹².
The simplified expression is 1024y¹⁴/x¹².