Question

1. the sum of 3 and \\(sqrt{2}\\) is

a. rational, since the sum can be expressed as an integer
b. rational, since the sum can be expressed as a nonterminating decimal
c. irrational, since the sum can be expressed as a terminating decimal
d. irrational, since the sum cannot be expressed as a terminating or repeating decimal

Explanation:

Step1: Classify the numbers

3 is a rational number (can be written as $\frac{3}{1}$), $\sqrt{2}$ is an irrational number (non-terminating, non-repeating decimal ~1.4142...).

Step2: Apply rational-irrational sum rule

The sum of a rational number and an irrational number is always irrational. An irrational number cannot be expressed as a terminating or repeating decimal.

Step3: Evaluate options

Options A and B claim the sum is rational, which is false. Option C incorrectly states the sum is a terminating decimal, which is not true for $3+\sqrt{2}$. Option D correctly identifies the sum as irrational with the right reasoning.

Answer:

D. Irrational, since the sum cannot be expressed as a terminating or repeating decimal