Solve. For Exercises 83 and 84, see the second Concept Check...

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check. A lottery prize of $\frac{15 x^{3}}{y^{2}}$ dollars is to be divided among $5 x$ people. Express the amount of money each person is to receive as a rational expression in $x$ and $y .$

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check.
A lottery prize of $\frac{15 x^{3}}{y^{2}}$ dollars is to be divided among $5 x$ people. Express the amount of money each person is to receive as a rational expression in $x$ and $y .$

Key Concepts

Rational Expressions

A rational expression is any fraction where both the numerator and the denominator are polynomials. Understanding how to work with these expressions involves knowing how to perform operations like addition, multiplication, and division, as well as simplifying the expression by canceling common factors, always minding that the original conditions (e.g., denominators not equal to zero) remain satisfied.

Division of Algebraic Expressions

Dividing one algebraic expression by another typically involves multiplying by the reciprocal of the divisor. This method allows for the conversion of division problems into multiplication problems, facilitating the simplification process and helping to manage the distribution of a quantity among several parts.

Simplification of Expressions

Simplification involves rewriting an expression in its most reduced form by canceling common factors, rearranging terms, and applying algebraic rules. This process is crucial to arrive at a clear, concise, and simplified answer that is easier to interpret and use in further computations.

Properties of Exponents

The properties of exponents, such as the product rule, quotient rule, and power rule, are fundamental when working with algebraic expressions that include powers. These properties help in correctly manipulating and combining terms with exponents, which is key in both simplifying expressions and solving algebraic problems.

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