Irrational numbers

Step 1: (6<\sqrt{t}<7) means (36<t<49). Step 2: (42) lies in this interval, so (\sqrt{42}) lies between (6) and (7). Step 3: Square the boundary numbers to understand square-root intervals.

Step 1: Multiply numerator and denominator by (\sqrt{2}). Step 2: (\frac{4}{\sqrt{2}}=\frac{4\sqrt{2}}{2}=2\sqrt{2}). Step 3: After rationalising, simplify the answer completely.

Step 1: First multiply the numbers inside the roots. Step 2: The first three give (144), (324), and (900), which are perfect squares; the fourth gives (\sqrt{22}). Step 3: After multiplication, check whether the resulting number is a perfect square.

Step 1: (\sqrt{128}=8\sqrt{2}), (\sqrt{72}=6\sqrt{2}), and (\sqrt{50}=5\sqrt{2}). Step 2: (8\sqrt{2}+6\sqrt{2}-5\sqrt{2}=9\sqrt{2}). Step 3: Convert all radicals into like form before adding or subtracting.

Step 1: Subtracting an irrational number from a rational number leaves an irrational part. Step 2: If the result were rational, then (q=p-(p-q)) would be rational, which is impossible. Step 3: Remember the rules for addition and subtraction of rational and irrational numbers.

Step 1: (\sqrt{c}\times\sqrt{12}=\sqrt{12c}). Step 2: (\sqrt{12c}=18), so (12c=324) and (c=27). Step 3: In square-root equations, square both sides to solve.

Step 1: Since (9<13<16), (\sqrt{13}) lies between (3) and (4). Step 2: (\sqrt{13}\approx3.606), so (3.61) is the closest. Step 3: In approximation, first set the range using perfect squares.

Step 1: (\sqrt{7}) and (\sqrt{7}) are both irrational. Step 2: Their difference is (0), which is rational. Step 3: The difference of equal irrational terms can be rational.

Step 1: (\sqrt{48}\times\sqrt{75}=\sqrt{3600}). Step 2: (\sqrt{3600}=60), so the result is rational. Step 3: In multiplication, multiply the inside numbers and check for a perfect square.

Step 1: In (\frac{1}{3-\sqrt{8}}), the conjugate of the denominator is (3+\sqrt{8}). Step 2: The denominator becomes (9-8=1), so the value is (3+\sqrt{8}). Step 3: Rationalising with the conjugate quickly simplifies the denominator.

Step 1: (6) is rational and (\sqrt{5}) is irrational. Step 2: The sum of a rational and an irrational number is irrational. Step 3: Adding an integer does not remove the irrational square-root part.

Step 1: (\sqrt{363}=11\sqrt{3}) and (\sqrt{75}=5\sqrt{3}). Step 2: (11\sqrt{3}-5\sqrt{3}=6\sqrt{3}). Step 3: Simplify radicals completely before subtracting.

Step 1: (\sqrt{24}) is irrational because (24) is not a perfect square. Step 2: Since (16<24<25), (4<\sqrt{24}<5). Step 3: In condition-based questions, check both irrationality and range.

Step 1: This is of the form \((a-b)(a+b)=a^2-b^2\). Step 2: \((\sqrt{11})^2-(\sqrt{2})^2=11-2=9\). Step 3: In conjugate multiplication, directly use the difference of squares.

Step 1: Rationalise (\frac{1}{2-\sqrt{3}}) by multiplying by (2+\sqrt{3}). Step 2: The denominator becomes (4-3=1), so the value is (2+\sqrt{3}). Step 3: Use rationalisation to identify equivalent forms.

Step 1: Multiply the given approximate value by (4). Step 2: (4\sqrt{3}\approx4\times1.732=6.928). Step 3: In approximation questions, directly use the given value.

Step 1: (6\sqrt{2}=\sqrt{36}\sqrt{2}). Step 2: This equals (\sqrt{72}). Step 3: To move the outside coefficient inside, multiply by its square.

Step 1: (\sqrt{27}=3\sqrt{3}). Step 2: (x=\sqrt{3}+3\sqrt{3}=4\sqrt{3}). Step 3: Simplify radicals before adding them.

Step 1: (\sqrt{6}\approx2.45) and (\sqrt{10}\approx3.16). Step 2: (3) lies between these two values. Step 3: Use approximate values or squaring to compare.

Step 1: (\sqrt{20}=2\sqrt{5}) and (\sqrt{45}=3\sqrt{5}). Step 2: The total is (6\sqrt{5}), which is irrational. Step 3: Simplify an expression before deciding its nature.

Step 1: Dividing by a non-zero rational number does not remove irrationality. Step 2: For example, (\frac{\sqrt{3}}{4}) remains irrational. Step 3: The condition (k\neq0) is necessary because division by zero is not possible.

Step 1: (\sqrt{216}=6\sqrt{6}) and (\sqrt{54}=3\sqrt{6}). Step 2: (6\sqrt{6}+3\sqrt{6}=9\sqrt{6}). Step 3: Add radicals only after they become like radicals.

Step 1: (4) is a non-zero rational number and (\sqrt{13}) is irrational. Step 2: (4\sqrt{13}) remains irrational. Step 3: Multiplication by zero is a special case, so focus on non-zero rational factors.

Step 1: If (\sqrt{a}=12), square both sides. Step 2: (a=144), and (144) is a perfect square. Step 3: Square both sides in square-root equations to find the original number.

Step 1: (\sqrt{75}=5\sqrt{3}), so (\sqrt{3}+\sqrt{75}=6\sqrt{3}). Step 2: (6\sqrt{3}) is irrational, so it is not rational. Step 3: Simplify each option before deciding its nature.