Irrational numbers

Step 1: An irrational number cannot be written as (\frac{p}{q}). Step 2: (\sqrt{2}) is not the square root of a perfect square, so it is irrational. Step 3: If the number under a square root is not a perfect square, it is usually irrational.

Step 1: The square root of a perfect square is rational. Step 2: (9) is a perfect square and (\sqrt{9}=3), so it is rational. Step 3: In square root questions, first check whether the number inside is a perfect square.

Step 1: (16) is a perfect square. Step 2: (\sqrt{16}=4), and (4) can be written as (\frac{4}{1}). Step 3: A number that can be written as a fraction is rational.

Step 1: (10) is not a perfect square. Step 2: So (\sqrt{10}) cannot be written exactly as (\frac{p}{q}). Step 3: Be careful while identifying square roots of non-perfect squares.

Step 1: (0.75), fractions, and integers are rational. Step 2: (6) is not a perfect square, so (\sqrt{6}) is irrational. Step 3: In options, eliminate clear rational numbers first.

Step 1: A rational number has either a terminating or recurring decimal form. Step 2: An irrational number has a non-terminating and non-recurring decimal form. Step 3: While judging decimals, carefully check recurring and non-recurring patterns.

Step 1: In (0.3333\ldots), the digit (3) repeats. Step 2: A recurring decimal is rational. Step 3: Do not call a repeating decimal irrational.

Step 1: The digits do not repeat in a fixed recurring pattern. Step 2: It is non-terminating and non-recurring, so it is irrational. Step 3: In long decimals, check whether there is a fixed repeating block.

Step 1: Multiplying an irrational number by a non-zero rational number keeps it irrational. Step 2: For example, (2 \times \sqrt{3}=2\sqrt{3}), which is irrational. Step 3: The non-zero condition is important because multiplication by (0) gives (0).

Step 1: (\sqrt{2}) is irrational and (3) is rational. Step 2: Adding a rational number does not make it rational. Step 3: Adding a simple rational number to an irrational number generally keeps it irrational.

Step 1: (\sqrt{2}+\sqrt{2}=2\sqrt{2}). Step 2: (\sqrt{2}) is irrational and (2) is non-zero rational, so (2\sqrt{2}) is irrational. Step 3: First simplify like radicals, then decide the type.

Step 1: Multiplying a square root by itself gives the number inside the root. Step 2: (\sqrt{5}\times\sqrt{5}=5), and (5) is rational. Step 3: The product of two irrational numbers is not always irrational.

Step 1: (\sqrt{7}\times\sqrt{7}=7). Step 2: Because there is one negative sign, the product is (-7). Step 3: Pay attention to both the sign and the square root.

Step 1: The question asks for the number that is not irrational, so look for a rational option. Step 2: (\sqrt{25}=5), so it is rational. Step 3: Be careful with negative wording in MCQs.

Step 1: Write (12=4 \times 3). Step 2: (\sqrt{12}=\sqrt{4 \times 3}=2\sqrt{3}). Step 3: While simplifying square roots, take the perfect square factor outside.

Step 1: (18=9 \times 2). Step 2: (\sqrt{18}=\sqrt{9 \times 2}=3\sqrt{2}). Step 3: Choosing the largest perfect square factor makes simplification faster.

Step 1: (3) is not a perfect square. Step 2: (\sqrt{3}) is irrational, so its decimal expansion is non-terminating and non-recurring. Step 3: An irrational decimal has no fixed repeating pattern.

Step 1: A rational number can be written in the form (\frac{p}{q}). Step 2: A number that cannot be written in this form is called irrational. Step 3: In definition questions, remember the condition (q \neq 0).

Step 1: Square roots of perfect squares are integers. Step 2: For example, (\sqrt{36}=6), so it is rational. Step 3: Do not assume every square root is irrational.

Step 1: (\sqrt{2}\times\sqrt{8}=\sqrt{16}). Step 2: (\sqrt{16}=4), so the value is rational. Step 3: While multiplying square roots, multiply the numbers inside the roots.

Step 1: Write (50=25 \times 2). Step 2: (\sqrt{50}=\sqrt{25 \times 2}=5\sqrt{2}). Step 3: A larger perfect square factor makes square-root simplification easier.

Step 1: (2) is rational and (\sqrt{5}) is irrational. Step 2: The sum of a rational and an irrational number is irrational. Step 3: In such questions, identify the nature of both parts separately.

Step 1: Subtracting a number from itself gives (0). Step 2: (\sqrt{11}-\sqrt{11}=0), and (0) is rational. Step 3: The difference of two irrational numbers is not always irrational.

Step 1: (\sqrt{3}) is irrational. Step 2: Dividing it by the non-zero rational number (2) still gives an irrational number. Step 3: Multiplying or dividing an irrational number by a non-zero rational number generally keeps it irrational.

Step 1: (\sqrt{4}=2) and (\sqrt{9}=3). Step 2: Adding them gives (2+3=5). Step 3: First find the square roots of perfect squares separately.