Irrational numbers

\(0.333\ldots=\frac{1}{3}\), so it is a rational number even though its decimal expansion never ends; it repeats. In contrast, \(\sqrt{2}\) has a non-terminating, non-repeating decimal. Exam tip: check whether the decimal repeats.

( \sqrt{18} ) is irrational, so its decimal expansion is non-terminating non-recurring. Recurring decimals are rational.

( \sqrt{14} ) is irrational, and every irrational number is also real. Real numbers include both rational and irrational numbers.

(27) is not a perfect square, so ( \sqrt{27} ) is irrational. Distinguish perfect squares from non-perfect squares.

An irrational number cannot be expressed as
p/q
, where
q \ne 0
. Hence, its decimal expansion is non-terminating and non-repeating; a repeating decimal is rational. Exam tip: look for “non-repeating.”

(32) is not a perfect square, so ( \sqrt{32} ) is irrational. Check perfect squares when seeing a square root.

( \sqrt{26} ) is the square root of a non-perfect square, so it is irrational. ( \frac{22}{7} ) is rational, but ( \pi ) is not.

The decimal expansion of an irrational number is non-terminating non-recurring. This is a key exam identification.

The governing concept is classification through exact simplification. Write 0.09 as 9/100. Then √0.09 = √(9/100) = √9/√100 = 3/10 = 0.3. Since 3/10 is a ratio of integers with a non-zero denominator, it is rational; its decimal form also terminates. The value is positive and real, so it is neither non-real nor an irrational number. It is also not a non-terminating, non-recurring decimal. This example shows that a square root of a decimal is not automatically irrational: when the decimal is the square of a rational number, its square root remains rational. Therefore option B is correct.

An irrational number has a decimal expansion that neither ends nor repeats a fixed pattern, so A is correct. Terminating or repeating decimals are rational. Exam tip: remember “non-terminating, non-repeating.”

( \sqrt{4}=2 ) is rational and ( \sqrt{7} ) is irrational. Check the type of both numbers in a pair.

( \sqrt{5} ) lies approximately between (2) and (3) and is irrational. A square root of a non-perfect square is a good example.

Adding a rational number to an irrational number gives an irrational number. (3) is rational and ( \sqrt{2} ) is irrational.

Multiplying non-zero rational (5) by irrational ( \sqrt{3} ) gives an irrational number. The zero case is different.

A rational number is a number that can be written as a fraction of two integers with a nonzero denominator. Every integer is rational, so if the square root of a number is an integer, it is automatically rational. A perfect square has an integer square root, such as 1, 4, 9, 16, or 64. This makes the first choice always rational under its stated condition.

For example, \(\sqrt{64}=8=\frac{8}{1}\), so it is rational. By contrast, the square root of a non-perfect square is irrational, such as \(\sqrt{2}\). A multiple of \(\pi\) is not automatically rational, and a non-terminating, non-recurring decimal is irrational by definition. Therefore the square root of a perfect square, option A, is the correct choice.

( \sqrt{20} ) is irrational and all irrational numbers are real. Real but not rational means irrational.

( \sqrt{3} ) is irrational, so its decimal is non-terminating non-recurring. The decimal of an irrational number is never recurring.

(6), (10), and (11) are not perfect squares, so all three square roots are irrational. Check each number separately.

( \sqrt{2}\times\sqrt{2}=2 ), so it is rational. The product of two irrational numbers is not always irrational.

The governing rule is √(ab) = √a × √b, together with the extraction of perfect-square factors from a radical. Factor 28 as 4 × 7, where 4 is a perfect square. Therefore √28 = √(4 × 7) = √4 × √7 = 2√7. The factor 4 can leave the radical because its square root is the integer 2, whereas 7 remains under the radical because it has no square factor greater than 1. Option A is therefore the simplified form. Option B is not equivalent because (7√2)^2 = 98, option C has square 112, and option D has square 196. Squaring 2√7 gives 4 × 7 = 28, which directly verifies the answer and shows why the other choices are invalid.