a. The result is irrational.
b. The result is irrational.
c. The result is rational.
d. The result is rational.
In the given expressions:
a. The sum of 5 and √19 is not a rational number because √19 is an irrational number. Therefore, the result is an irrational number.
b. The sum of 3/7 and 9/5 is a rational number because both 3/7 and 9/5 are rational numbers. Therefore, the result is a rational number.
c. The product of √6 and 6 is not a rational number because √6 is an irrational number. Therefore, the result is an irrational number.
Now let's fill in the blanks:
a. The operation is addition. The first number is 3/4 (rational) and the second number is √3/4 (irrational). The result is the sum of these two numbers. So, the result is irrational.
b. The operation is addition. The first number is 2/3 (rational) and the second number is √2/3 (irrational). The result is the sum of these two numbers. So, the result is irrational.
c. The operation is multiplication. The first number is 3/4 (rational) and the second number is 3.6 (rational). The result is the product of these two numbers. So, the result is rational.
d. The operation is addition. The first number is 3/4 (rational) and the second number is 2.7 (rational). The result is the sum of these two numbers. So, the result is rational.
Regarding Carmen's error, she incorrectly assumes that 11.2 is not a rational number. However, 11.2 can be written as 112/10, which is a rational number. Therefore, the sum of 11.2 and 19 is a rational number.