Irrational numbers

Step 1: For (a=4,b=9), the left side is (2+3=5). Step 2: The right side is (\sqrt{13}), which is not (5). Step 3: When adding square roots, the numbers inside the roots are not added directly.

Step 1: (\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}). Step 2: For (a=50,b=2), it becomes (\sqrt{25}=5), which is rational, so it should not be selected. Step 3: For an irrational quotient, (\frac{a}{b}) should not be a perfect square; none of the listed options gives that.

Step 1: (\frac{\sqrt{18}}{\sqrt{5}}=\sqrt{\frac{18}{5}}). Step 2: (\frac{18}{5}) is not a perfect square of a rational number, so the result is irrational. Step 3: In quotients of radicals, check whether the fraction inside is a perfect square.

Step 1: (\frac{1}{2+\sqrt{3}}=2-\sqrt{3}). Step 2: (x+\frac{1}{x}=(2+\sqrt{3})+(2-\sqrt{3})=4). Step 3: Recognizing the conjugate reciprocal saves long calculation.

Step 1: (\sqrt{8}=2\sqrt{2}). Step 2: So (x=3+2\sqrt{2}), which contains an irrational part. Step 3: Do not combine rational and irrational terms into a single radical.

Step 1: Assume (\sqrt{2}+\sqrt{3}) is rational. Step 2: Squaring gives (5+2\sqrt{6}) rational, which would force (\sqrt{6}) to be rational, impossible. Step 3: Squaring is useful for sums of two different surds.

Step 1: (\frac{1}{\sqrt{5}+\sqrt{3}}=\frac{\sqrt{5}-\sqrt{3}}{2}). Step 2: Therefore (\frac{2}{x}=\sqrt{5}-\sqrt{3}). Step 3: (x-\frac{2}{x}=(\sqrt{5}+\sqrt{3})-(\sqrt{5}-\sqrt{3})=2\sqrt{3}).

Step 1: For (a=4), (\sqrt{4}=2). Step 2: For (b=18), (\sqrt{18}=3\sqrt{2}), which is irrational; so the sum (2+3\sqrt{2}) is irrational. Step 3: A rational plus an irrational remains irrational.

Step 1: (x^2=2+7+2\sqrt{14}=9+2\sqrt{14}). Step 2: Therefore (x^2-9=2\sqrt{14}), which is irrational. Step 3: Square first, then subtract the rational part.

Step 1: (12=4\times3). Step 2: (\sqrt{12}=\sqrt{4}\sqrt{3}=2\sqrt{3}). Step 3: Take the perfect square factor outside the radical.

Step 1: (x+1=\sqrt{3}). Step 2: Therefore ((x+1)^2=(\sqrt{3})^2=3). Step 3: Simplify the inner expression first, then square it.

Step 1: Both numbers are positive, so compare their squares. Step 2: ((2\sqrt{3})^2=12) and ((3\sqrt{2})^2=18), so (3\sqrt{2}) is greater. Step 3: Squaring is a safe method for comparing positive surds.

Step 1: (xy=2\sqrt{5}\times5\sqrt{2}=10\sqrt{10}). Step 2: (\sqrt{10}) is irrational, so (10\sqrt{10}) is irrational. Step 3: If the product inside the root is not a perfect square, the result may remain irrational.

Step 1: This decimal does not terminate. Step 2: The number of (1)'s keeps changing, so there is no fixed recurring block. Step 3: A non-terminating non-recurring decimal is irrational.

Step 1: Write (x^2-4x=x(x-4)). Step 2: With (x=\sqrt{13}+2), (x-4=\sqrt{13}-2), so the product is (13-4=9). Step 3: A conjugate form may be hidden in such expressions.

Step 1: Multiply by (2+\sqrt{3}) to rationalize the denominator. Step 2: (\frac{(2+\sqrt{3})^2}{4-3}=4+4\sqrt{3}+3=7+4\sqrt{3}). Step 3: When multiplying by the conjugate, the numerator may become a full square.

Step 1: ((2+\sqrt{3})^2=4+4\sqrt{3}+3). Step 2: This equals (7+4\sqrt{3}). Step 3: In such questions, identify the form (m+n+2\sqrt{mn}).

Step 1: (\sqrt{80}=4\sqrt{5}), (\sqrt{45}=3\sqrt{5}), and (\sqrt{20}=2\sqrt{5}). Step 2: (4\sqrt{5}-3\sqrt{5}+2\sqrt{5}=3\sqrt{5}), so none of the listed options is correct. Step 3: In such questions, trust your simplification before matching options.

Step 1: (\sqrt{80}=4\sqrt{5}), (\sqrt{45}=3\sqrt{5}), and (\sqrt{20}=2\sqrt{5}). Step 2: (4\sqrt{5}-3\sqrt{5}+2\sqrt{5}=3\sqrt{5}), which is irrational. Step 3: Handle the signs carefully when three terms are involved.

Step 1: To make (x+\sqrt{2}) rational, (x) should contain a (-\sqrt{2}) part. Step 2: If (x=3-\sqrt{2}), then (x+\sqrt{2}=3), which is rational. Step 3: Look for cancellation of the irrational part.