Real Numbers

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.

\(\sqrt{625}=25\) and \(\sqrt{361}=19\), since \(25^2=625\) and \(19^2=361\). Therefore, \(x=25-19=6\). Hence, 6 is the correct option. A value such as 5 can result from evaluating one of the square roots incorrectly. Exam tip: verify a square root quickly by squaring your answer.

Here, \(ab=\sqrt{45}\times\sqrt{20}=\sqrt{45\times20}=\sqrt{900}=30\). Alternatively, \(\sqrt{45}=3\sqrt{5}\) and \(\sqrt{20}=2\sqrt{5}\), so \(ab=(3\sqrt{5})(2\sqrt{5})=6\times5=30\). The value \(20\) is only \(b^2\), not \(ab\). Exam tip: while multiplying square roots, use \(\sqrt{x}\sqrt{y}=\sqrt{xy}\) and look for a perfect square.

Since \(8^2=64<70\), we have \(\sqrt{70}>8\). Also, \(9^2=81\), which lies between 70 and 90; therefore, \(\sqrt{70}<9<\sqrt{90}\). Hence, 9 is the correct number. While 8 is less than \(\sqrt{70}\), 10 is greater than \(\sqrt{90}\). Exam tip: To find an integer between square roots, compare the square of that integer with the given numbers.

Since \(507=169\times3=13^2\times3\), \(\sqrt{507}=\sqrt{13^2\times3}=13\sqrt{3}\). Therefore, \(13\sqrt{3}\) is correct. For example, \(12\sqrt{3}\) squares to \(432\), not \(507\). Exam tip: To simplify a square root, first identify the greatest perfect-square factor of the number.

\(\sqrt{441}=21\) and \(\sqrt{324}=18\). Therefore, \(x=21+18=39\), so option C is correct. Option 38 may result from calculating one of the square roots incorrectly. In exams, evaluate each perfect square root separately before adding.

Since \(1200=400\times 3\), and \(400\) is a perfect square, \(\sqrt{1200}=\sqrt{400\times3}=\sqrt{400}\sqrt{3}=20\sqrt{3}\). The option \(10\sqrt{3}\) has only half the required coefficient, so it is incorrect. Exam tip: to simplify a square root, first identify the greatest perfect-square factor of the number.

Since \(9^2=81\) and \(10^2=100\), \(\sqrt{95}\) lies between 9 and 10. Its approximate value is \(9.75\), which is closer to 10 than to 9. Therefore, 10 is the correct answer. Although 9 is nearby, its distance from \(\sqrt{95}\) is about 0.75, while the distance from 10 is about 0.25. Exam tip: Compare the nearby perfect squares to estimate the nearest integer to a square root.

\(1452=484\times 3=22^2\times 3\). Therefore, \(\sqrt{1452}=\sqrt{22^2\times 3}=22\sqrt{3}\). Hence, \(22\sqrt{3}\) is correct. \(11\sqrt{3}\) is incorrect because the largest square factor is \(22^2\), not \(11^2\). Exam tip: To simplify a surd, first identify the greatest perfect-square factor of the number inside the square root.

\(\sqrt{20}=\sqrt{4\times5}=2\sqrt{5}\). Therefore, \(\sqrt{5}+\sqrt{20}=\sqrt{5}+2\sqrt{5}=3\sqrt{5}\). Option C is only the simplified form of \(\sqrt{20}\) and omits the original \(\sqrt{5}\). Exam tip: add the coefficients of like surd terms, using \(a\sqrt{b}+c\sqrt{b}=(a+c)\sqrt{b}\).

Since \(3^2=9<11<15<16=4^2\), we get \(3<\sqrt{11}<\sqrt{15}<4\). Both square roots lie between 3 and 4, and there is no whole number between them. Neither 3 nor 4 lies between the two square roots, so option C is correct. Exam tip: compare nearby perfect squares to locate square roots quickly.

To rationalise the denominator, multiply the numerator and denominator by the conjugate \(\sqrt{7}-\sqrt{5}\). The denominator becomes \((\sqrt{7}+\sqrt{5})(\sqrt{7}-\sqrt{5})=7-5=2\). Therefore, the result is \(\frac{\sqrt{7}-\sqrt{5}}{2}\). Option C omits division by 2. Exam tip: for a denominator that is a sum of two surds, use the conjugate with a minus sign.

Since \(80<81=9^2\), and the square-root function increases for non-negative numbers, \(\sqrt{80}<\sqrt{81}=9\). Hence, option C is correct. Option B may look close, but \(\sqrt{80}\) is not equal to \(\sqrt{81}\). Exam tip: compare square roots by comparing the numbers with nearby perfect squares.

Since \(4^2=16<18<19<25=5^2\), we have \(4<\sqrt{18}<\sqrt{19}<5\). Both square roots lie between 4 and 5, and there is no whole number strictly between 4 and 5. Hence, no whole number lies between the two square roots, so option C is correct. Options A and B are incorrect because 4 is less than \(\sqrt{18}\), while 5 is greater than \(\sqrt{19}\). Exam tip: use the nearest perfect squares to locate square roots on the number line.