Irrational numbers

Step 1: An irrational number cannot be written exactly in the form (\frac{p}{q}). Step 2: (14) is not a perfect square, so (\sqrt{14}) is irrational. Step 3: In square-root questions, first check whether the number inside is a perfect square.

Step 1: (121) is a perfect square. Step 2: (\sqrt{121}=11), and (11) can be written as (\frac{11}{1}). Step 3: Square roots of perfect squares are rational.

Step 1: Terminating and recurring decimals are rational. Step 2: (0.01001000100001\ldots) has no fixed repeating pattern, so it is non-terminating and non-recurring. Step 3: For decimals, check the repetition pattern, not just the length.

Step 1: Write (108=36 \times 3). Step 2: (\sqrt{108}=\sqrt{36 \times 3}=6\sqrt{3}). Step 3: While simplifying a square root, choosing the largest perfect square factor is helpful.

Step 1: (\sqrt{7}) is irrational and (4) is rational. Step 2: Adding a rational number does not remove the irrational part. Step 3: The sum of a rational and an irrational number is generally irrational.

Step 1: In multiplication of square roots, multiply the numbers inside. Step 2: (\sqrt{2}\times\sqrt{50}=\sqrt{100}=10). Step 3: The product of two irrational numbers can sometimes be rational.

Step 1: To find the rational option, look for the square root of a perfect square. Step 2: (144) is a perfect square and (\sqrt{144}=12). Step 3: Recognising perfect squares makes such questions very quick.

Step 1: (\sqrt{27}=3\sqrt{3}). Step 2: (\sqrt{3}+3\sqrt{3}=4\sqrt{3}). Step 3: Before adding, simplify radicals and make them like terms.

Step 1: (\frac{5}{6}) is a rational number. Step 2: Its decimal form is (0.8333\ldots), which is recurring. Step 3: Rational numbers have decimal expansions that are either terminating or recurring.

Step 1: Both terms have the same square root. Step 2: (\sqrt{11}+\sqrt{11}=2\sqrt{11}). Step 3: For like radicals, add only the coefficients, not the numbers inside the roots.

Step 1: Subtracting a number from itself gives (0). Step 2: So (\sqrt{19}-\sqrt{19}=0), which is rational. Step 3: Ordinary subtraction rules also apply to irrational terms.

Step 1: (\sqrt{3}\times\sqrt{75}=\sqrt{225}). Step 2: (\sqrt{225}=15), so the result is rational. Step 3: If the number inside becomes a perfect square after multiplication, the answer can be rational.

Step 1: Square roots of perfect squares are integers. Step 2: For example, (\sqrt{49}=7), so it is rational. Step 3: In statement-based questions, using an example helps check the truth.

Step 1: (242=121 \times 2). Step 2: (\sqrt{242}=\sqrt{121 \times 2}=11\sqrt{2}). Step 3: Recognising large perfect squares like (121) is very useful in simplification.

Step 1: (\sqrt{45}=3\sqrt{5}). Step 2: (\sqrt{5}+3\sqrt{5}=4\sqrt{5}). Step 3: While adding radicals, simplify them until like radicals appear.

Step 1: Since (36<45<49), (6<\sqrt{45}<7). Step 2: (45) is not a perfect square, so (\sqrt{45}) is irrational. Step 3: In interval questions, use nearby perfect squares.

Step 1: (\sqrt{7}) is irrational. Step 2: Dividing it by the non-zero rational number (3) keeps the result irrational. Step 3: Division by a non-zero rational number does not remove irrationality.

Step 1: (8) is rational. Step 2: (\sqrt{15}) is irrational because (15) is not a perfect square. Step 3: In pair questions, check the nature of both numbers separately.

Step 1: Write (175=25 \times 7). Step 2: (\sqrt{175}=\sqrt{25 \times 7}=5\sqrt{7}). Step 3: Take the perfect square factor outside and keep the remaining number inside.

Step 1: Four like radical terms are being added. Step 2: (\sqrt{10}+\sqrt{10}+\sqrt{10}+\sqrt{10}=4\sqrt{10}). Step 3: When adding like radicals, add only the coefficients.