Irrational numbers

Step 1: (\sqrt{27}=3\sqrt{3}) and (\sqrt{12}=2\sqrt{3}). Step 2: The sum is (3\sqrt{3}+2\sqrt{3}=5\sqrt{3}), which is irrational. Step 3: Do not combine separate square roots as (\sqrt{39}).

Step 1: Coprime means (p) and (q) have no common factor except (1). Step 2: In the proof of (\sqrt{2}), both (p) and (q) turn out even, so they have common factor (2). Step 3: This contradiction proves that (\sqrt{2}) is not rational.

Step 1: (\sqrt{32}=4\sqrt{2}). Step 2: (\sqrt{32}-\sqrt{2}=4\sqrt{2}-\sqrt{2}=3\sqrt{2}), which is irrational. Step 3: For like surds, subtract only the coefficients.

Step 1: (\sqrt{44}=\sqrt{4\times11}=2\sqrt{11}). Step 2: Hence (x=\sqrt{11}+2\sqrt{11}=3\sqrt{11}), and (\sqrt{11}) is irrational. Step 3: For like surds, add only the coefficients, not the numbers inside the roots.

Step 1: A non-terminating decimal can still be rational if it is recurring. Step 2: In (0.37373737\ldots), the block (37) repeats, so it is rational. Step 3: Do not call a decimal irrational just because it is non-terminating; check for a repeating block.

Step 1: (a) and (b) are conjugates. Step 2: (ab=(\sqrt{3})^2-2^2=3-4=-1), which is rational and negative. Step 3: In conjugate multiplication, the middle irrational terms cancel.

Step 1: Simplify each radical first. Step 2: (\sqrt{75}=5\sqrt{3}), so (\sqrt{75}-4\sqrt{3}=\sqrt{3}), which is irrational. Step 3: Options where like terms cancel completely may give rational zero.

Step 1: If (k) is a perfect square, then (\sqrt{k}) is rational and the fraction becomes rational. Step 2: (18) is not a perfect square, so (\sqrt{18}) is irrational and (\frac{5}{\sqrt{18}}) remains irrational. Step 3: In such questions, first check whether (k) is a perfect square.

Step 1: (\sqrt{5}) and (\sqrt{20}=2\sqrt{5}) are both irrational and different. Step 2: (\frac{\sqrt{5}}{\sqrt{20}}=\frac{\sqrt{5}}{2\sqrt{5}}=\frac{1}{2}), which is rational. Step 3: A common irrational factor can cancel in a quotient.

Step 1: (x-4=(4+\sqrt{6})-4). Step 2: On simplifying, (x-4=\sqrt{6}), and since (6) is not a perfect square, (\sqrt{6}) is irrational. Step 3: When rational terms cancel, check the nature of the remaining radical.

Step 1: Dividing by a non-zero rational number is the same as multiplying by its reciprocal. Step 2: (\frac{1}{r}) is also a non-zero rational number, so (\frac{x}{r}) remains irrational. Step 3: In division questions, always check that the denominator is not zero.

Step 1: (\sqrt{18}=3\sqrt{2}). Step 2: (\sqrt{2}+\sqrt{18}=\sqrt{2}+3\sqrt{2}=4\sqrt{2}), which is irrational. Step 3: For like surds, add only the outside coefficients.

Step 1: Terminating and recurring decimals are rational. Step 2: (1.01001000100001\ldots) is non-terminating and has no fixed repeating block. Step 3: To identify an irrational decimal, check both non-termination and non-repetition.

Step 1: Substitute the given value of (x). Step 2: (x-\sqrt{3}=(\sqrt{3}+2)-\sqrt{3}=2), which is rational. Step 3: Like irrational terms may cancel, so decide the nature only after simplifying.

Step 1: (\frac{\sqrt{45}}{3}=\frac{3\sqrt{5}}{3}=\sqrt{5}). Step 2: Since (5) is not a perfect square, (\sqrt{5}) is irrational. Step 3: Do not choose an answer in multiplication or division of surds without simplifying.

Step 1: The square root of a positive integer is rational only when the integer is a perfect square. Step 2: (72) is not a perfect square, so (\sqrt{72}) is irrational. Step 3: Being even does not make a square root rational.

Step 1: This is multiplication of conjugates. Step 2: (ab=5^2-(\sqrt{7})^2=25-7=18). Step 3: In conjugate multiplication, the middle irrational terms cancel.

Step 1: (3=\sqrt{9}) and (4=\sqrt{16}). Step 2: (10) is not a perfect square and (9<10<16), so (\sqrt{10}) is an irrational number between (3) and (4). Step 3: A non-perfect square between two square numbers helps find an irrational number between two integers.

Step 1: If (\frac{1}{x}) is rational and non-zero, then its reciprocal is also rational. Step 2: Therefore (x) is rational, not irrational. Step 3: In reciprocal questions, always check the non-zero condition.

Step 1: The conjugate of the denominator is (2-\sqrt{5}). Step 2: (\frac{3}{2+\sqrt{5}}\times\frac{2-\sqrt{5}}{2-\sqrt{5}}=\frac{3(2-\sqrt{5})}{4-5}=3(\sqrt{5}-2)). Step 3: Use the difference of squares in the denominator when multiplying by a conjugate.