Irrational numbers

Step 1: (12=4\times3). Step 2: (\sqrt{12}=2\sqrt{3}), and (\sqrt{3}) is irrational. Step 3: After simplification, if a non-square remains inside the root, the number stays irrational.

Step 1: A negative sign changes direction, not rationality. Step 2: If (-x) were rational, then (x) would also be rational, which is false. Step 3: Treat the sign of a number and its type separately.

Step 1: A rational number can be written as (\frac{p}{q}), where (p,q) are integers and (q\neq0). Step 2: An irrational number cannot be written in that form. Step 3: In definition questions, always check the condition (q\neq0).

Step 1: (\frac{7}{8}) and (4.25) are terminating decimals. Step 2: (\frac{2}{3}) is non-terminating recurring. (\sqrt{17}) is irrational, so its decimal expansion is non-terminating and non-recurring. Step 3: Quickly identify square roots of non-perfect squares.

Step 1: Use ((a+b)^2=a^2+2ab+b^2). Step 2: (x^2=2+2\sqrt{6}+3=5+2\sqrt{6}), which has an irrational part. Step 3: Do not forget the middle term when squaring a sum of surds.

Step 1: Multiply numerator and denominator by (\sqrt{3}). Step 2: (\frac{1}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{3}}{3}). Step 3: Rationalizing the denominator gives a cleaner exam answer.

Step 1: (3) is rational and (\sqrt{2}) is irrational. Step 2: The sum of a rational and an irrational number is irrational. Step 3: Adding an integer does not remove the irrational nature of the surd.

Step 1: (0\times\sqrt{7}=0), (\sqrt{7}\times\sqrt{7}=7), and (\sqrt{28}\div\sqrt{7}=2) are rational. Step 2: (4\sqrt{7}) is a non-zero rational multiple of an irrational number, so it is irrational. Step 3: Quickly identify multiplication by zero as rational.

Step 1: A positive integer has a rational square root only when it is a perfect square. Step 2: For example, (\sqrt{16}=4), but (\sqrt{18}) is irrational. Step 3: Check perfect squares to decide the nature of a square root.

Step 1: In the proof for (\sqrt{2}), we assume (\sqrt{2}=\frac{a}{b}). Step 2: This gives (a^2=2b^2), so (a^2) is even and hence (a) is even. Step 3: This parity argument leads to a contradiction.

Step 1: (\sqrt{13}) is irrational. Step 2: Its square is ((\sqrt{13})^2=13), which is rational. Step 3: The square of an irrational number is not always irrational, so examine examples carefully.

Step 1: (72=36\times2). Step 2: (\sqrt{72}=\sqrt{36}\sqrt{2}=6\sqrt{2}), which is irrational. Step 3: Use the largest perfect square factor for quick simplification.

Step 1: The denominator has (\sqrt{5}), which is irrational. Step 2: Rationalizing gives (x=\frac{2\sqrt{5}}{5}), a non-zero rational multiple of an irrational number. Step 3: Rationalizing the denominator often reveals the number type clearly.

Step 1: First look for like irrational terms. Step 2: ((2+\sqrt{3})+(5-\sqrt{3})=7) because (\sqrt{3}) and (-\sqrt{3}) cancel. Step 3: Opposite irrational terms can produce a rational result.

Step 1: From (x^2=7), (x=\sqrt{7}) or (x=-\sqrt{7}). Step 2: Among the options, (\sqrt{7}) is present and it is irrational. Step 3: Remember both positive and negative roots, then match the given options.

Step 1: (\sqrt{9}=3) and (\sqrt{16}=4). Step 2: Their sum is (7), which is rational. Step 3: If both radicands are perfect squares, the sum is easily rational.

Step 1: (3) is rational and (\sqrt{2}) is irrational. Step 2: A rational number minus an irrational number remains irrational. Step 3: Do not classify the whole expression by looking only at the rational part.

Step 1: This is a product of conjugates. Step 2: ((1+\sqrt{2})(1-\sqrt{2})=1-(\sqrt{2})^2=1-2=-1). Step 3: In conjugate multiplication, the middle irrational terms cancel.

Step 1: In (0.123123123\ldots), the block (123) repeats. Step 2: A recurring decimal is rational. Step 3: Do not call a decimal irrational just because it is non-terminating; check repetition.

Step 1: The sum of two irrational numbers can be rational. Step 2: For example, (\sqrt{2}+(-\sqrt{2})=0). Therefore, saying (a+b) is always irrational is false. Step 3: Be careful with universal statements about two irrational numbers.