Question

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Assume a and b are nonzero rational numbers and c and d are irrational numbers. For each of the following expressions, determine whether the result is irrational, rational, or both. Justify your answers.
Part A: b(a + d)
Part B: left parenthesis b square root of five end square root right parenthesis cubed
Part C: a + (cb)

Answer 1

Part A: Well product of two rationals is always rational. And the sum of a rational and an irrational is irrational.

They give you too much freedom in your choice though. Like for part A, what if we let b=-1 (rational) c=pi (irrational d=pi (irrational) Then bc+d = -pi + pi = 0 (rational) Maybe they didn't want you to consider 0 because otherwise all of these options can be rational.

Do you understand the difference between a rational and irrational number? Rationals can be written as a fraction containing whole numbers. Irrationals can not. Here are some examples of rational numbers: 3/2 is rational because its a whole number divided by a whole number. 7 is a rational number because it can be written as 7/1. −4 is a rational number, it also can be written in this form −4/1. This next one maybe seem surprising, but, 0 is a rational number. It also can be written as 0/1, but not as 0/0 (Which is not a number).


So in the end 
A:  virtually any sum or product involving an irrational number will be irrational. 
B. (√7)^2 = 7 ... rational 
C. all numbers involved are rational, so this result is rational

Answer 2

All results are irrational.
You have the following "rules."
- the sum of rational and irrational is irrational
- the product of rational and irrational is irrational

A. The first rule applies inside parentheses. The second rule applies then.

B. The result is (rational)*(5^(3/2)). The latter factor is irrational, so the second rule applies.

C. The second rule applies inside parentheses. The first rule applies then.