Real Numbers

Correct answer: C. \(0.101001000100001\ldots\)

Explanation: In C, the zeros between successive 1s keep increasing, so no fixed block repeats. It is non-terminating and non-repeating, hence irrational. Exam tip: every recurring decimal is rational.

Correct answer: C. 20

Explanation: \(\sqrt{400}\) is the positive number whose square is 400. Since \(20 \times 20 = 400\), \(\sqrt{400}=20\). For example, the square of 10 is 100, so 10 is not correct. Exam tip: for a perfect square, check which number gives the given value when multiplied by itself.

Correct answer: A. \(3\sqrt{5}\)

Explanation: Since \(45=9\times5\), and 9 is the greatest perfect-square factor of 45, \(\sqrt{45}=\sqrt{9\times5}=\sqrt9\times\sqrt5=3\sqrt5\). The expression \(5\sqrt3\) would require \(45=25\times3\), which is not true. Exam tip: To simplify a square root, first identify the greatest perfect-square factor of the number.

Correct answer: B. 1

Explanation: Given \(x=\sqrt{36}-\sqrt{25}\). Since \(\sqrt{36}=6\) and \(\sqrt{25}=5\), \(x=6-5=1\). Option 0 would result only if the two square roots were equal, which they are not. Exam tip: evaluate each perfect-square root before subtracting.

Correct answer: B. 18

Explanation: \(\sqrt{81}=9\) and \(\sqrt{4}=2\). Therefore, \(\sqrt{81}\times\sqrt{4}=9\times2=18\), so option B is correct. Getting 16 would result from multiplying incorrect square-root values. Exam tip: For perfect squares such as 81 and 4, evaluate each square root separately first.

Correct answer: B. 7.5

Explanation: \(\sqrt{50}\approx 7.07\) and \(\sqrt{64}=8\). Therefore, the required number must be greater than 7.07 and less than 8. \(7.5\) satisfies this condition. 7 is less than \(\sqrt{50}\), while 8 is equal to the upper bound, so neither lies between them. Exam tip: use nearby perfect squares to estimate square roots quickly.

Correct answer: B. \(9\sqrt{3}\)

Explanation: Since \(243=81\times3=9^2\times3\), \(\sqrt{243}=\sqrt{9^2\times3}=9\sqrt{3}\). The original distractor \(3\sqrt{27}\) also equals \(9\sqrt3\), so it would create a second correct answer; the options have therefore been revised to keep one correct answer. Exam tip: identify the greatest perfect-square factor before simplifying a square root.

Correct answer: C. 4

Explanation: Here, \(ab=\sqrt{2}\times\sqrt{8}=\sqrt{2\times8}=\sqrt{16}=4\). Therefore, the correct answer is 4. \(\sqrt{10}\) results from an incorrect addition-based approach, whereas the given quantities are being multiplied. Exam tip: for multiplication of square roots, use \(\sqrt{x}\,\sqrt{y}=\sqrt{xy}\).

Correct answer: C. \(\sqrt{17}\)

Explanation: Since \(4^2=16\) and \(5^2=25\), \(\sqrt{17}\) lies between 4 and 5. As \(17>16\), \(\sqrt{17}>\sqrt{16}=4\), so \(\sqrt{17}\) is the greatest number. Also, \(\sqrt{15}<4\), and 3.8 is less than 4. Exam tip: compare square roots using nearby perfect squares.

Correct answer: C. 18

Explanation: Since \(18 \times 18 = 324\), \(\sqrt{324}=18\). The principal square root is always non-negative, so it is not \(-18\). Exam tip: Memorising the squares of numbers from 15 to 20 helps solve such questions quickly.

Correct answer: B. 3

Explanation: Since \(8<9<10\), taking positive square roots gives \(\sqrt{8}<\sqrt{9}<\sqrt{10}\). As \(\sqrt{9}=3\), the number 3 lies between the two given numbers. 2.5 is less than \(\sqrt{8}\), whereas 3.5 and 4 are greater than \(\sqrt{10}\). Exam tip: To locate an integer between square roots, compare the squares of nearby integers.

Correct answer: C. 4

Explanation: \(\sqrt{100}=10\) and \(\sqrt{36}=6\). Therefore, \(x=10-6=4\). Option 5 can result from an incorrect subtraction. Exam tip: evaluate each perfect-square root first, then perform the operation.

Correct answer: C. 17

Explanation: Since \(17 \times 17 = 289\), \(\sqrt{289}=17\). The square of 16 is 256, so 16 is not correct. Exam tip: Memorising square roots of perfect squares such as 256, 289, and 324 helps solve such questions quickly.

Correct answer: B. 2

Explanation: \(\frac{a}{b}=\frac{\sqrt{12}}{\sqrt{3}}=\sqrt{\frac{12}{3}}=\sqrt{4}=2\). Therefore, the correct option is 2. Option 4 results from incorrectly treating \(\sqrt{4}\) as 4. Exam tip: when dividing square roots, divide the numbers inside the roots first and then simplify the square root.

Correct answer: B. 25

Explanation: The principal square root of 625 is the positive number whose square is 625. Since \(25 \times 25 = 625\), \(\sqrt{625}=25\). The squares of 20 and 30 are 400 and 900 respectively, so they cannot be correct. Exam tip: verify a perfect-square root by squaring the selected number.

Correct answer: B. 5

Explanation: Since \(5^2=25\) and \(6^2=36\), \(\sqrt{26}\) lies between 5 and 6. As 26 is much closer to 25, \(\sqrt{26}\approx 5.10\), which is closest to 5. It is not 6 because its distance from \(\sqrt{26}\) is about 0.90. Exam tip: Compare the given number with nearby perfect squares to find the nearest integer square root.

Correct answer: B. \(6\sqrt{5}\)

Explanation: \(180=36\times5\), and \(36\) is a perfect square. Therefore, \(\sqrt{180}=\sqrt{36\times5}=\sqrt{36}\sqrt{5}=6\sqrt{5}\). If the result were \(3\sqrt{5}\), its square would be \(45\), not 180. Exam tip: To simplify a square root, first identify the greatest perfect-square factor of the number.

Correct answer: C. 18

Explanation: \(\sqrt{49}=7\) and \(\sqrt{121}=11\). Therefore, \(x=7+11=18\), so option C is correct. Choosing 17 would be incorrect because the sum of the two square roots is 18. Exam tip: evaluate square roots of perfect squares separately before adding or subtracting them.

Correct answer: C. 12

Explanation: For positive numbers, \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\). Hence, \(\sqrt{48}\times\sqrt{3}=\sqrt{48\times3}=\sqrt{144}=12\). Therefore, the correct answer is 12. Although 13 is close, \(13^2=169\), not 144. Exam tip: When multiplying square roots, multiply the numbers inside the roots first and then identify a perfect square.

Correct answer: A. \(10\sqrt{5}\)

Explanation: Since \(500=100\times 5\), and \(100\) is a perfect square, \(\sqrt{500}=\sqrt{100\times5}=\sqrt{100}\sqrt{5}=10\sqrt{5}\). The option \(5\sqrt{10}\) is not equal to \(\sqrt{500}\), as its square is \(250\). Exam tip: To simplify a square root, first identify the greatest perfect-square factor of the number.

Correct answer: C. 4

Explanation: \(\sqrt{196}=14\) and \(\sqrt{100}=10\), since \(14^2=196\) and \(10^2=100\). Therefore, \(x=14-10=4\). Option 5 can result from an incorrect subtraction or an incorrect square-root value. Exam tip: evaluate each perfect-square root separately before performing the operation.

Correct answer: B. 3.5

Explanation: 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.

Correct answer: C. \(4\sqrt{2}\)

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

Correct answer: C. 12

Explanation: \(\sqrt{225}=15\), \(\sqrt{25}=5\), 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.

Correct answer: A. \(5\sqrt{3}\)

Explanation: 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.