Irrational numbers

Step 1: The decimal of an irrational number does not terminate. Step 2: It also does not repeat in a fixed pattern. Step 3: Decimal expansion is a useful way to identify irrational numbers.

Step 1: (49) is a perfect square, so (\sqrt{49}=7) is rational. Step 2: (50) is not a perfect square, so (\sqrt{50}) is irrational. Step 3: Even for nearby numbers, check perfect squares carefully.

Step 1: (\sqrt{8}=2\sqrt{2}). Step 2: (\sqrt{2}+2\sqrt{2}=3\sqrt{2}). Step 3: Simplify radicals before adding them.

Step 1: Write (27=9 \times 3). Step 2: (\sqrt{27}=\sqrt{9 \times 3}=3\sqrt{3}). Step 3: When (9) appears as a factor inside a square root, take it out as (3).

Step 1: Irrational numbers are not integers because integers can be written in the form (\frac{p}{q}). Step 2: So saying every irrational number is an integer is false. Step 3: In false-statement questions, check every option separately.

Step 1: (45=9 \times 5). Step 2: (\sqrt{45}=\sqrt{9 \times 5}=3\sqrt{5}). Step 3: Identifying the perfect square factor is the key to simplifying square roots.

Step 1: We are given (x=\sqrt{2}). Step 2: (x^2=(\sqrt{2})^2=2). Step 3: Squaring a square root gives the number inside it.

\(\sqrt{3}\times\sqrt{12}=\sqrt{3\times12}=\sqrt{36}=6\). Hence, the correct answer is 6. \(\sqrt{15}\) would result only if the product of the numbers under the radicals were 15, which is not the case here. Exam tip: for positive numbers, use \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\).

Step 1: In (\sqrt{5}) and (\sqrt{7}), the numbers inside the roots are not perfect squares. Step 2: Hence both are irrational. Step 3: In pair questions, check both numbers, not just one.

Step 1: (20=4 \times 5). Step 2: (\sqrt{20}=\sqrt{4 \times 5}=2\sqrt{5}). Step 3: Take the perfect square outside the root and leave the remaining factor inside.

Step 1: (\sqrt{2}) is irrational. Step 2: (\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}), which is irrational. Step 3: When a square root is in the denominator, rationalising helps identify the number.

Step 1: Adding a rational number does not remove the irrational part. Step 2: For example, (4+\sqrt{2}) is irrational. Step 3: The sum of a rational and an irrational number is an important rule to remember.

Step 1: Both (\sqrt{2}) and (\sqrt{3}) are irrational. Step 2: Their sum is not (\sqrt{5}); (\sqrt{2}+\sqrt{3}) remains irrational. Step 3: Do not add the numbers inside different square roots directly.

Step 1: While multiplying square roots, multiply the numbers inside. Step 2: (\sqrt{2}\times\sqrt{3}=\sqrt{6}), which is irrational. Step 3: In multiplication, multiply the inside numbers; do not add them.

Step 1: Write (72=36 \times 2). Step 2: (\sqrt{72}=\sqrt{36 \times 2}=6\sqrt{2}). Step 3: Taking the largest perfect square factor gives a cleaner final form.

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.