Real Numbers

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.

Since \(588=196\times 3=14^2\times 3\), \(\sqrt{588}=\sqrt{14^2\times 3}=14\sqrt{3}\). \(13\sqrt{3}\) is not correct because \(13^2\times3=507\), not 588. Exam tip: To simplify a square root, first identify the greatest perfect-square factor of the number.

\(\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}\) and \(\sqrt{8}=\sqrt{4\times2}=2\sqrt{2}\). Therefore, \(a-b=5\sqrt{2}-2\sqrt{2}=3\sqrt{2}\). \(5\sqrt{2}\) is only the value of \(a\), not the difference. Exam tip: Simplify surds to the same radical form before subtracting them.

\(968=484\times2=22^2\times2\). Therefore, \(\sqrt{968}=\sqrt{22^2\times2}=22\sqrt{2}\). Although \(21\sqrt{2}\) may seem close, \(21^2\times2=882\), not 968. Exam tip: to simplify a square root, identify the greatest perfect-square factor of the number.

Since \(529=23^2\), we get \(\sqrt{529}=23\). Therefore, \(x=\frac{23}{\sqrt{23}}=\frac{\sqrt{23}\times\sqrt{23}}{\sqrt{23}}=\sqrt{23}\). Option 23 is incorrect because division by \(\sqrt{23}\) still remains. Exam tip: Use \(a=\sqrt{a}\times\sqrt{a}\) to simplify fractions involving square roots.

Since \(726=121\times6=11^2\times6\), \(\sqrt{726}=\sqrt{11^2\times6}=11\sqrt{6}\). The option \(11\sqrt{7}\) is incorrect because its radicand would be \(121\times7=847\). Exam tip: to simplify a square root, first identify the greatest perfect-square factor of the number.

Since \(288=144\times 2=12^2\times 2\), \(\sqrt{288}=\sqrt{12^2\times 2}=12\sqrt{2}\). Option A, \(6\sqrt{8}\), has the same value but is not in simplest form because \(\sqrt{8}\) can be simplified further. Exam tip: to simplify a surd, identify the greatest perfect-square factor of the number.

Since \(6^2=36\) and \(7^2=49\), \(\sqrt{41}\) lies between 6 and 7. Also, \(6.5^2=42.25\) and \(41<42.25\), so \(\sqrt{41}<6.5\). Therefore, \(\sqrt{41}\) is closer to 6 than to 7; its value is about 6.40. Exam tip: Compare the squares of consecutive integers to quickly identify the nearest integer to a square root.

\(\sqrt{32}=\sqrt{16\times2}=4\sqrt{2}\) and \(\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}\). Therefore, \(a+b=4\sqrt{2}+3\sqrt{2}=7\sqrt{2}\). \(6\sqrt{2}\) would result only if the coefficients added to 6; here they are 4 and 3. Exam tip: simplify surds into like radical terms before adding them.

\(847=49\times17=7^2\times17\). Therefore, \(\sqrt{847}=\sqrt{7^2\times17}=7\sqrt{17}\). Since \(17\) is not a perfect square, the expression cannot be simplified further. Also, \(7\sqrt{121}=77\), so it is not equal to \(\sqrt{847}\). Exam tip: To simplify a square root, identify the greatest perfect-square factor of the number.