Real Numbers

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.

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.

\(\sqrt{81}=9\) 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.

\(\sqrt{50}\approx 7.07\) 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.

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.

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 \(\sqrt{x}\,\sqrt{y}=\sqrt{xy}\).

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.

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.

The governing concept is the classification of rational and irrational numbers. A rational number can be written as p/q, where p and q are integers and q is nonzero. In option D, √5 × √5 = (√5)^2 = 5, and every integer is rational because it can be written as 5/1. The other expressions remain irrational: √2 + √3 is irrational, √7 + 2 is irrational because adding a rational number to an irrational number remains irrational, and π + 1 is irrational for the same reason. Therefore option D is the only rational expression. The multiplication sign matters: √5 + √5 would equal 2√5, which is irrational, but the given product simplifies exactly to 5.

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.

\(\sqrt{100}=10\) 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.

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.

\(\frac{a}{b}=\frac{\sqrt{12}}{\sqrt{3}}=\sqrt{\frac{12}{3}}=\sqrt{4}=2\). 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.

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.

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.

\(180=36\times5\), 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.

\(\sqrt{49}=7\) 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.

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.

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.

\(\sqrt{196}=14\) 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.