Irrational numbers

Step 1: Subtracting a number from itself gives zero. Step 2: (\sqrt{7}-\sqrt{7}=0), and (0) is rational. Step 3: Normal subtraction rules also apply to irrational numbers.

Step 1: (\sqrt{2}\times\sqrt{5}=\sqrt{10}). Step 2: (10) is not a perfect square, so (\sqrt{10}) is irrational. Step 3: After multiplication, check the number inside the new root.

Step 1: If (n) is not a perfect square, (\sqrt{n}) is not rational. Step 2: For example, (\sqrt{6}) is irrational. Step 3: Testing a statement with an example makes it easier to judge.

Step 1: (98=49 \times 2). Step 2: (\sqrt{98}=\sqrt{49 \times 2}=7\sqrt{2}). Step 3: Recognising larger perfect squares like (49) helps in simplification.

Step 1: (\sqrt{12}=2\sqrt{3}). Step 2: (\sqrt{3}+2\sqrt{3}=3\sqrt{3}). Step 3: Before adding, convert radicals into like terms if possible.

Step 1: Since (16<17<25), (4<\sqrt{17}<5). Step 2: (17) is not a perfect square, so (\sqrt{17}) is irrational. Step 3: Use nearby perfect squares in interval questions.

Step 1: (\sqrt{5}) is irrational. Step 2: Dividing it by the non-zero rational number (5) keeps it irrational. Step 3: A rational denominator alone does not make the whole expression rational.

Step 1: (3) is rational. Step 2: (\sqrt{6}) is irrational because (6) is not a perfect square. Step 3: In a pair, identify each number separately.

Step 1: Write (125=25 \times 5). Step 2: (\sqrt{125}=\sqrt{25 \times 5}=5\sqrt{5}). Step 3: Take the perfect square (25) outside as (5).

Step 1: Three like irrational terms are being added. Step 2: (\sqrt{6}+\sqrt{6}+\sqrt{6}=3\sqrt{6}). Step 3: Count like radicals as coefficients.

Step 1: (\sqrt{45}=3\sqrt{5}) and (\sqrt{20}=2\sqrt{5}). Step 2: (3\sqrt{5}-2\sqrt{5}=\sqrt{5}). Step 3: Simplify both radicals before subtracting.

Step 1: ((\sqrt{2})^2=2). Step 2: (2) is rational because it can be written as (\frac{2}{1}). Step 3: The square of an irrational number can sometimes be rational.

Step 1: (x=\sqrt{5}) is irrational. Step 2: (5x=5\sqrt{5}), and (5) is a non-zero rational number. Step 3: Multiplying an irrational number by a non-zero rational number keeps it irrational.

Step 1: The question asks for the number that is not irrational, so find the rational one. Step 2: (\sqrt{36}=6), which is rational. Step 3: Read negative wording carefully in MCQs.

Step 1: (\sqrt{8}=2\sqrt{2}) and (\sqrt{18}=3\sqrt{2}). Step 2: (2\sqrt{2}+3\sqrt{2}=5\sqrt{2}). Step 3: Radicals can be added only after they become like radicals.

Step 1: (\sqrt{49}=7). Step 2: (7) is rational, so saying (\sqrt{49}) is irrational is false. Step 3: While choosing a false statement, check perfect squares carefully.

Step 1: Write (150=25 \times 6). Step 2: (\sqrt{150}=\sqrt{25 \times 6}=5\sqrt{6}). Step 3: Take the perfect square factor outside and leave the remaining factor inside.

Step 1: (\sqrt{5}\times\sqrt{20}=\sqrt{100}). Step 2: (\sqrt{100}=10), which is rational. Step 3: In multiplication, first multiply the numbers inside the roots.

Step 1: Since (25<30<36), (5<\sqrt{30}<6). Step 2: (30) is not a perfect square, so (\sqrt{30}) is irrational. Step 3: Use nearby perfect squares to locate a square root.

Step 1: (200=100 \times 2). Step 2: (\sqrt{200}=\sqrt{100 \times 2}=10\sqrt{2}). Step 3: Recognising a large perfect square like (100) gives the answer quickly.