Real Numbers

\(\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}\) 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.

Since \(432=144\times 3=12^2\times 3\), \(\sqrt{432}=\sqrt{12^2\times 3}=12\sqrt{3}\). Therefore, option A is correct. For example, \(13\sqrt{3}\) is incorrect because its square is \(507\), not \(432\). Exam tip: To simplify a square root, first identify the greatest perfect-square factor of the number.

\(\sqrt{8}=\sqrt{4\times2}=2\sqrt{2}\) and \(\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}\). Therefore, \(a+b=2\sqrt{2}+3\sqrt{2}=5\sqrt{2}\). \(4\sqrt{2}\) can result from simplifying the second surd incorrectly. Exam tip: First take out perfect-square factors from surds, then add terms with the same radical part.

\(\sqrt{225}=15\) and \(\sqrt{144}=12\). Therefore, \(x=15-12=3\). Option 27 results from incorrectly subtracting 144 from 225; the question requires subtracting the square roots. Exam tip: evaluate each perfect square root separately before performing the subtraction.

\(\sqrt{500}=\sqrt{100\times5}=10\sqrt{5}\) and \(\sqrt{125}=\sqrt{25\times5}=5\sqrt{5}\). Therefore, \(\sqrt{500}-\sqrt{125}=10\sqrt{5}-5\sqrt{5}=5\sqrt{5}\). \(10\sqrt{5}\) is only the value of \(\sqrt{500}\), not the result after subtraction. Exam tip: before subtracting surds, extract perfect-square factors from each radicand.

\(\sqrt{324}=18\) because \(18^2=324\), and \(\sqrt{36}=6\) because \(6^2=36\). Therefore, \(x=18+6=24\), so option C is correct. An answer such as 23 may result from evaluating a square root incorrectly. Exam tip: find each square root of a perfect square separately before adding.

\(675=225\times3=15^2\times3\), so \(\sqrt{675}=\sqrt{15^2\times3}=15\sqrt{3}\). Choosing \(15\sqrt{5}\) would give a radicand of \(225\times5=1125\), not 675. Exam tip: identify the greatest perfect-square factor before simplifying a surd.

\(\sqrt{27}=\sqrt{9\times3}=3\sqrt{3}\) and \(\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}\). Therefore, \(a-b=3\sqrt{3}-2\sqrt{3}=\sqrt{3}\), so option A is correct. \(2\sqrt{3}\) is only the value of \(b\), not the difference. Exam tip: simplify surds to the same radical form before subtracting them.

The set of real numbers contains every rational number and every irrational number. Thus, \(\mathbb{R}=\mathbb{Q}\cup(\mathbb{R}\setminus\mathbb{Q})\), so option A is correct. Integers and natural numbers are only subsets of rational numbers, so they do not represent all real numbers. Exam tip: for the real-number set, check that both rational and irrational numbers are included.

\(882=441\times2=21^2\times2\). Therefore, \(\sqrt{882}=\sqrt{21^2\times2}=21\sqrt{2}\). Hence, option C is correct. For example, \(20\sqrt{2}\) squares to \(800\), so it cannot equal \(\sqrt{882}\). Exam tip: To simplify a square root, identify the greatest perfect-square factor of the number.

\(\sqrt{400}=20\) and \(\sqrt{225}=15\). Therefore, \(x=20-15=5\). Hence, 5 is the correct option. Getting 4 would indicate an error in evaluating the square roots or subtracting them. Exam tip: evaluate the square roots of perfect squares separately before subtracting.

Since \(8^2=64\) and \(9^2=81\), \(\sqrt{80}\) lies between 8 and 9 and is very close to 9. In fact, \(\sqrt{80}\approx 8.94\), so 9 is the closest number. Option 8 is not the closest because 80 is much nearer to 81 than to 64. Exam tip: Compare nearby perfect squares to estimate a square root quickly.