Irrational numbers

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.

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.