Irrational numbers

Step 1: The product of radicals is (\sqrt{2}\cdot \sqrt{3}=\sqrt{6}). Step 2: Since (6) is not a perfect square (\sqrt{6}) is irrational. Step 3: In multiplication the numbers inside radicals multiply, not add.

Step 1: (\sqrt{9}=3). Step 2: (4+3=7), a rational number. Step 3: Do not call every expression with a square root irrational immediately.

Step 1: From (r+s=s), subtract (s) from both sides to get (r=0). Step 2: (0) is rational, so the condition is possible. Step 3: Form a simple equation before judging number types.

Step 1: (\sqrt{98}=7\sqrt{2}) and (\sqrt{50}=5\sqrt{2}). Step 2: The difference is (2\sqrt{2}), which is irrational. Step 3: Directly subtracting numbers inside radicals is wrong.

Step 1: (\sqrt{2}+\sqrt{18}=4\sqrt{2}). Step 2: (\sqrt{8}+\sqrt{12}=2\sqrt{2}+2\sqrt{3}). Since (\sqrt{3}>\sqrt{2}), the second expression is greater. Step 3: Simplify first and compare carefully.

Step 1: In square-root questions first check whether the number inside is a perfect square. Step 2: A perfect square may give a rational square root while a non-perfect square often gives an irrational value. Step 3: Simplifying is the safest first step in identification questions.

Step 1: Since (28=4\cdot 7), (\sqrt{28}=2\sqrt{7}). Step 2: Now (\sqrt{7}+2\sqrt{7}=3\sqrt{7}), and (\sqrt{7}) is irrational. Step 3: In exams, combine like radicals by adding their coefficients.

Step 1: (\frac{3}{2}) is rational and (\sqrt{5}) is irrational. Step 2: If their sum were rational, then (\sqrt{5}) would become the difference of two rational numbers, which is impossible. Step 3: For rational-plus-irrational questions, contradiction is a very useful method.

Step 1: The first option is a product of conjugate terms. Step 2: ((\sqrt{6}+\sqrt{2})(\sqrt{6}-\sqrt{2})=6-2=4), which is rational. Step 3: Identifying conjugates helps remove radicals quickly.

Step 1: The square root of a rational fraction is rational when both numerator and denominator can be perfect squares. Step 2: For example, (\sqrt{\frac{4}{9}}=\frac{2}{3}), so a ratio of two perfect squares is a safe condition. Step 3: Being positive or less than (1) does not guarantee a rational square root.