Irrational numbers

Step 1: (7<\sqrt{t}<8) means (49<t<64). Step 2: (57) lies in this interval, so (\sqrt{57}) lies between (7) and (8). Step 3: Square the boundary numbers to understand square-root intervals.

Step 1: Multiply numerator and denominator by (\sqrt{3}). Step 2: (\frac{6}{\sqrt{3}}=\frac{6\sqrt{3}}{3}=2\sqrt{3}). Step 3: After rationalising, simplify the answer completely.

Step 1: First multiply the numbers inside the roots. Step 2: The first three give (400), (900), and (784), which are perfect squares; the fourth gives (\sqrt{30}). Step 3: After multiplication, check whether the resulting number is a perfect square.

Step 1: (\sqrt{242}=11\sqrt{2}), (\sqrt{98}=7\sqrt{2}), and (\sqrt{32}=4\sqrt{2}). Step 2: (11\sqrt{2}+7\sqrt{2}-4\sqrt{2}=14\sqrt{2}). Step 3: Convert all radicals into like form before adding or subtracting.

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

Step 1: (\sqrt{d}\times\sqrt{20}=\sqrt{20d}). Step 2: (\sqrt{20d}=30), so (20d=900) and (d=45). Step 3: In square-root equations, square both sides to solve.

Step 1: Since (16<19<25), (\sqrt{19}) lies between (4) and (5). Step 2: (\sqrt{19}\approx4.359), so (4.36) is the closest. Step 3: In approximation, first set the range using perfect squares.

Step 1: (\sqrt{17}) and (\sqrt{17}) 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{63}\times\sqrt{112}=\sqrt{7056}). Step 2: (\sqrt{7056}=84), so the result is rational. Step 3: In multiplication, multiply the inside numbers and check for a perfect square.

Step 1: In (\frac{1}{4-\sqrt{15}}), the conjugate of the denominator is (4+\sqrt{15}). Step 2: The denominator becomes (16-15=1), so the value is (4+\sqrt{15}). Step 3: Rationalising with the conjugate quickly simplifies the denominator.

Step 1: (8) is rational and (\sqrt{11}) 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{507}=13\sqrt{3}) and (\sqrt{192}=8\sqrt{3}). Step 2: (13\sqrt{3}-8\sqrt{3}=5\sqrt{3}). Step 3: Simplify radicals completely before subtracting.

Step 1: (\sqrt{40}) is irrational because (40) is not a perfect square. Step 2: But (36<40<49), so (\sqrt{40}) is greater than (6); the other options are rational. Step 3: This option set has no valid answer and should be corrected.

Step 1: This is of the form \((a-b)(a+b)=a^2-b^2\). Step 2: \((\sqrt{13})^2-(\sqrt{5})^2=13-5=8\). Step 3: In conjugate multiplication, directly use the difference of squares.

Step 1: Rationalise (\frac{4}{3-\sqrt{5}}) by multiplying by (3+\sqrt{5}). Step 2: The denominator becomes (9-5=4), so the value is (3+\sqrt{5}). Step 3: Use rationalisation to identify equivalent forms.

Step 1: Multiply the given approximate value by (5). Step 2: (5\sqrt{5}\approx5\times2.236=11.180). Step 3: In approximation questions, directly use the given value.

Step 1: (7\sqrt{2}=\sqrt{49}\sqrt{2}). Step 2: This equals (\sqrt{98}). Step 3: To move the outside coefficient inside, multiply by its square.

Step 1: (\sqrt{45}=3\sqrt{5}). Step 2: (x=\sqrt{5}+3\sqrt{5}=4\sqrt{5}). Step 3: Simplify radicals before adding them.

Step 1: (\sqrt{15}\approx3.87) and (\sqrt{20}\approx4.47). Step 2: (4) lies between these two values. Step 3: Use approximate values or squaring to compare.

Step 1: (\sqrt{28}=2\sqrt{7}) and (\sqrt{63}=3\sqrt{7}). Step 2: The total is (6\sqrt{7}), 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{384}=8\sqrt{6}) and (\sqrt{54}=3\sqrt{6}). Step 2: (8\sqrt{6}+3\sqrt{6}=11\sqrt{6}). Step 3: Add radicals only after they become like radicals.

Step 1: (6) is a non-zero rational number and (\sqrt{19}) is irrational. Step 2: (6\sqrt{19}) remains irrational. Step 3: Multiplication by zero is a special case, so focus on non-zero rational factors.

Step 1: If (\sqrt{a}=15), square both sides. Step 2: (a=225), and (225) is a perfect square. Step 3: Square both sides in square-root equations to find the original number.

Step 1: (\sqrt{98}=7\sqrt{2}), so (\sqrt{2}+\sqrt{98}=8\sqrt{2}). Step 2: (8\sqrt{2}) is irrational, so it is not rational. Step 3: Simplify each option before deciding its nature.