Real Numbers

Since \(3^2=9<11\) and \(4^2=16>13\), both \(\sqrt{11}\) and \(\sqrt{13}\) lie between 3 and 4. More precisely, \(\sqrt{11}\approx 3.32\) and \(\sqrt{13}\approx 3.61\), so 3.5 lies between them. The number 3 is less than \(\sqrt{11}\), while 4 and 4.5 are greater than \(\sqrt{13}\). Exam tip: use nearby perfect squares to compare square roots quickly.

\(\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}\). Therefore, \(a+b=3\sqrt{2}+\sqrt{2}=4\sqrt{2}\), so option C is correct. \(3\sqrt{2}\) is only the value of \(a\), not the sum. Exam tip: simplify surds to the same radicand before adding them.

The governing rule is to factor the radicand so that a perfect-square factor can be taken outside the square root. Since 363 = 121 × 3 and 121 = 11², we have √363 = √(11² × 3) = √(11²)√3 = 11√3. The principal square root is positive, so the outside factor is +11. Therefore option B is correct. A reliable check is to square the result: (11√3)² = 121 × 3 = 363. Option A does not preserve the correct factorization, while options C and D introduce incorrect extra factors. The expression is fully simplified because 3 has no perfect-square factor greater than 1, so no further extraction is possible.

\(\sqrt{225}=15\), \(\sqrt{25}=5\), and \(\sqrt{64}=8\). Therefore, \(x=15+5-8=12\), so 12 is the correct option. A value such as 13 can result from an error in handling the subtraction. Exam tip: evaluate each square root first, then perform addition and subtraction carefully.

The governing rule is √(ab) = √a × √b for nonnegative factors, together with extracting the largest perfect-square factor. Factor 192 as 64 × 3, where 64 = 8². Therefore √192 = √(64 × 3) = √64 × √3 = 8√3, so option B is the fully simplified form. Although 4√12 has the same value, it is not completely simplified because 12 still contains the square factor 4: 4√12 = 4(2√3) = 8√3. Thus the wording “simplified form” makes B the intended unique answer. Options C and D are numerically wrong because their squares are 768 and 432, respectively, rather than 192.

Since \(12=4\times3\) and \(27=9\times3\), \(\sqrt{12}=2\sqrt{3}\) and \(\sqrt{27}=3\sqrt{3}\). Therefore, \(x=2\sqrt{3}+3\sqrt{3}=5\sqrt{3}\). The expression \(3\sqrt{3}\) is only the value of \(\sqrt{27}\), not the sum of both terms. Exam tip: extract perfect-square factors from surds before combining like surds.

\(\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}\) and \(\sqrt{8}=\sqrt{4\times2}=2\sqrt{2}\). Hence, \(a=3\sqrt{2}-2\sqrt{2}=\sqrt{2}\), so option A is correct. \(2\sqrt{2}\) is only the simplified form of \(\sqrt{8}\), not of the difference. Exam tip: simplify surds to forms with the same radicand before subtracting them.

For positive numbers, \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\). Therefore, \(x=\sqrt{2}\times\sqrt{50}=\sqrt{100}=10\). Hence, option B is correct. Exam tip: multiply the numbers inside the square roots first and then look for a perfect square.

\(\sqrt{20}=\sqrt{4\times5}=2\sqrt{5}\). Therefore, \(a+b=\sqrt{5}+2\sqrt{5}=3\sqrt{5}\), so option A is correct. \(4\sqrt{5}\) would result from incorrectly simplifying \(\sqrt{20}\) as \(3\sqrt{5}\). Exam tip: simplify each surd by taking out perfect-square factors before adding like surds.

For positive numbers, \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\). Hence, \(\sqrt{7}\times\sqrt{63}=\sqrt{7\times63}=\sqrt{441}=21\). Therefore, 21 is the correct option. Although 22 is close, \(22^2=484\), not 441. Exam tip: When multiplying square roots, first combine the radicands under one square root.

When \(n\) is a natural number that is not a perfect square, \(\sqrt{n}\) cannot be expressed as a ratio of two integers, so it is irrational. Merely being an integer is not enough: for example, if \(n=4\), then \(\sqrt{4}=2\), which is rational. Exam tip: The square root of a natural number is rational only when the number is a perfect square.

\(\sqrt{98}=\sqrt{49\times2}=7\sqrt{2}\) and \(\sqrt{32}=\sqrt{16\times2}=4\sqrt{2}\). Hence, \(\sqrt{98}-\sqrt{32}=7\sqrt{2}-4\sqrt{2}=3\sqrt{2}\). A choice such as \(2\sqrt{2}\) can result from subtracting the coefficients incorrectly. Exam tip: before subtracting surds, first extract perfect-square factors from each radicand.

Write \(54=9\times6\), where \(9\) is a perfect square. Hence, \(\sqrt{54}=\sqrt{9\times6}=\sqrt9\times\sqrt6=3\sqrt6\). For example, \(2\sqrt6\) is incorrect because its square is \(24\), not 54. Exam tip: To simplify a square root, identify the greatest perfect-square factor of the number.

Since \(\sqrt{12}=2\sqrt{3}\), we get \(ab=\sqrt{3}\times2\sqrt{3}=2\times3=6\). The option \(2\sqrt{3}\) is only the simplified value of \(b\), not of \(ab\). Exam tip: While multiplying surds, use \(\sqrt{x}\cdot\sqrt{y}=\sqrt{xy}\), or simplify each surd first.

\(\sqrt{169}=13\) and \(\sqrt{25}=5\). Therefore, \(x=13+5=18\). The value 17 may result from evaluating one of the square roots incorrectly. Exam tip: first find each square root of a perfect square, then add the results.

\(320=64\times 5\), and \(64\) is the greatest perfect-square factor. Therefore, \(\sqrt{320}=\sqrt{64\times5}=\sqrt{64}\sqrt{5}=8\sqrt{5}\). In option A, \(\sqrt{20}\) can be simplified further, so it is not in simplest form. Exam tip: To simplify a square root, identify the greatest perfect-square factor of the number.

\(\sqrt{144}=12\) and \(\sqrt{196}=14\). Therefore, \(x=12+14=26\). Option 24 may result from incorrectly evaluating one of the square roots. Exam tip: recognise perfect squares such as \(144=12^2\) and \(196=14^2\) before adding their square roots.

Since \(20<25<30\) and \(25=5^2\), we get \(\sqrt{20}<\sqrt{25}=5<\sqrt{30}\). Therefore, 5 lies between the two numbers. For comparison, \(4^2=16\), which is less than 20, while \(6^2=36\), which is greater than 30. Exam tip: To test an integer between square roots, compare its square with the numbers inside the roots.

Since \(147=49\times3\) and \(75=25\times3\), \(\sqrt{147}=7\sqrt{3}\) and \(\sqrt{75}=5\sqrt{3}\). Therefore, \(7\sqrt{3}+5\sqrt{3}=12\sqrt{3}\), so the correct answer is \(12\sqrt{3}\). A choice such as \(11\sqrt{3}\) may result from adding the coefficients incorrectly. Exam tip: first extract perfect-square factors, then add the coefficients of like surds.

\(\sqrt{256}=16\) and \(\sqrt{16}=4\). Therefore, \(x=16\div4=4\), so option B is correct. \(8\) would result if the divisor were \(2\), but here \(\sqrt{16}=4\). Exam tip: evaluate each square root before dividing.