Final answer:
Expressions involving powers are evaluated using different rules: writing as a product of powers, multiplying exponents, and subtracting exponents. The first expression is written as a product of powers for x, the second involves multiplying the exponents of the base 5, and the third requires subtracting the exponents while dividing the bases.
Step-by-step explanation:
The expressions given correspond to different methods used in manipulating powers according to the rules of exponents. Here's how you would match each expression to the correct method: 4x³.⁵: To evaluate this expression, write it as a product of powers. In mathematical terms, this means recognizing that a fractional exponent indicates a root, so you translate x³.⁵ as x³ times the square root of x or x³ ∙ x¹⁄₂. 5³.⁵: This expression requires you to multiply the exponents. The power of a power rule states that when you raise a number to an exponent and that result is again raised to another exponent, you multiply the exponents. Hence, 5³.⁵ is equivalent to 5^(3∙5) which simplifies to 5⁴¹₅. 7² ÷ 3²: To evaluate this expression, you subtract the exponents. This is based on the rule that when dividing like bases, you subtract the exponents of the terms, so 7² ÷ 3² simplifies to 7/3 all raised to the power of 2.