Final answer:
Rational expressions are closed under addition, subtraction, multiplication, and division. The operations can be performed by finding a common denominator, multiplying the numerators and denominators, or multiplying by the reciprocal.
Step-by-step explanation:
Addition:
Rational expressions are closed under addition if the sum of two rational expressions is also a rational expression. To determine if this is true, we can use the fact that rational expressions are closed under addition if their denominators are the same. We can add the rational expressions by finding a common denominator, then adding the numerators. For example, if we have (3/x) + (2/x), the common denominator is x and the sum is (3+2)/x = 5/x.
Subtraction:
Rational expressions are closed under subtraction if the difference of two rational expressions is also a rational expression. To determine if this is true, we can use the same method as addition, but subtract the numerators instead. For example, if we have (3/x) - (2/x), the common denominator is x and the difference is (3-2)/x = 1/x.
Multiplication:
Rational expressions are closed under multiplication if the product of two rational expressions is also a rational expression. To multiply rational expressions, we can multiply the numerators and denominators. For example, if we have (3/x) * (2/y), the product is (3*2)/(x*y) = 6/(x*y).
Division:
Rational expressions are closed under division if the quotient of two rational expressions is also a rational expression. To divide rational expressions, we can multiply the first expression by the reciprocal of the second expression. For example, if we have (3/x) / (2/y), we can rewrite it as (3/x) * (y/2). Then, we multiply the numerators and denominators to get (3y)/(2x). Therefore, rational expressions are closed under all of the given operations.