Irrational numbers

Step 1: Since (9<10<16), we get (3<\sqrt{10}<4). Step 2: (10) is not a perfect square, so (\sqrt{10}) is irrational. Step 3: Use nearby perfect squares to locate square roots.

Step 1: (\sqrt{3}) is irrational. Step 2: (2) is a non-zero rational number, so (2\sqrt{3}) remains irrational. Step 3: Do not call a radical rational just because it has a coefficient outside.

Step 1: (32=16 \times 2). Step 2: (\sqrt{32}=\sqrt{16 \times 2}=4\sqrt{2}). Step 3: To fully simplify the answer, take out the largest perfect square.

Step 1: The decimal value of (\sqrt{2}) is about (1.414). Step 2: Among the given options, (1.41) is the closest. Step 3: Remembering approximate values of common square roots helps in estimation.

Step 1: The value of (\sqrt{3}) is approximately (1.732). Step 2: So (1.73) is the closest option. Step 3: In estimation questions, first see between which perfect squares the number lies.

Step 1: The value of (\sqrt{5}) is about (2.236). Step 2: (2.24) is the closest value. Step 3: While choosing nearest values, do not depend only on integer boundaries.

Step 1: (\sqrt{2}\approx 1.414) and (\sqrt{3}\approx 1.732). Step 2: (1.5) lies between these two values. Step 3: Approximate values help in comparing square roots.

Step 1: (64) is a perfect square. Step 2: (\sqrt{64}=8), so it is rational. Step 3: Square roots of larger perfect squares can also be rational.

Step 1: The square root of a perfect square is an integer. Step 2: If (n) is not a perfect square, (\sqrt{n}) is not rational. Step 3: This rule is very useful in Class 10 irrational number questions.

Step 1: Write (75=25 \times 3). Step 2: (\sqrt{75}=\sqrt{25 \times 3}=5\sqrt{3}). Step 3: While simplifying a square root, split the inside number into a perfect square and the remaining factor.

Step 1: An irrational number cannot be written exactly as a fraction. Step 2: (15) is not a perfect square, so (\sqrt{15}) is irrational. Step 3: For square-root options, first check whether the number inside is a perfect square.

Step 1: (81) is a perfect square. Step 2: (\sqrt{81}=9), and (9) can be written as (\frac{9}{1}). Step 3: The square root of a perfect square is always rational.

Step 1: Terminating or recurring decimals are rational. Step 2: (0.1234567891011\ldots) has no fixed repeating pattern, so it is non-terminating and non-recurring. Step 3: To identify an irrational decimal, check for a repeating rule.

Step 1: Write (28=4 \times 7). Step 2: (\sqrt{28}=\sqrt{4 \times 7}=2\sqrt{7}). Step 3: While simplifying a square root, take the perfect square factor outside.

Step 1: When multiplying square roots, multiply the numbers inside. Step 2: (\sqrt{2}\times\sqrt{18}=\sqrt{36}=6). 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: (100) is a perfect square and (\sqrt{100}=10). Step 3: In (\sqrt{n}), if (n) is a perfect square, the value is rational.

Step 1: (\sqrt{2}) is irrational and (5) 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: (48=16 \times 3). Step 2: (\sqrt{48}=\sqrt{16 \times 3}=4\sqrt{3}). Step 3: Using the largest perfect square gives the simplest form.

Step 1: A number with a terminating decimal is rational. Step 2: (\frac{3}{8}=0.375), so its decimal terminates. Step 3: If the denominator has only factors (2) and (5), the decimal terminates.

Step 1: Like radicals are added like like terms. Step 2: (\sqrt{7}+\sqrt{7}=2\sqrt{7}). Step 3: In addition, do not add the numbers inside roots to write (\sqrt{14}).