Final Answer:
To transform the quantity [(5√x)^7]^3 into an expression with a rational exponent, you can follow these steps:
Simplify the nested exponent: (5√x)^7 = 5^(7) * (√x)^7 = 5^7 * x^(7/2).
Apply the exponent to the exponent: [(5√x)^7]^3 = (5^7 * x^(7/2))^3.
Use the distributive property of exponents: Expand the parentheses: (5^7)^3 * (x^(7/2))^3 = 5^(7*3) * x^(3 * (7/2)).
Simplify the expression: Combine the powers of 5 and x: 125 * x^(21/2).
Therefore, [(5√x)^7]^3 can be expressed as 125x^(21/2). This form has a rational exponent, making it easier to perform further calculations or analysis.