As a data analyst, you are tasked with migrating legacy manufacturing formulas into a new calculation engine that only supports rational (fractional) exponents instead of radical symbols. Recalling the rule for converting compound radicands to rational exponents, match each legacy radical expression to its correctly formatted equivalent for the new system.

An engineering technician is updating a database that only accepts formulas with rational exponents. They need to convert the expression $\sqrt{\left(\frac{5x}{8y}\right)^3}$ from a technical manual into the software. Which of the following is the correct rational exponent form for this expression?

Rational Exponent Formatting for Compound Bases

A database administrator is migrating legacy mathematical formulas into a new system that only supports rational exponents. When converting the radical expression $(\sqrt[3]{{}7x})^4$, the administrator enters it as ${}7x^{\frac{4}{3}}$. True or False: This is a mathematically correct representation of the original radical expression.

A technical writer is updating the digital manual for a precision machining tool that uses the formula $(\sqrt[3]{2x})^4$. When converting this radical expression into rational exponent form, the writer must enclose the base in ____ to ensure the fractional exponent applies to both the coefficient and the variable.