Real Numbers

Given \(x=\sqrt{2}\), \(3x-x=(3-1)x=2x=2\sqrt{2}\). The option \(\sqrt{2}\) is only the value of \(x\), whereas the expression asks for \(2x\). Exam tip: when adding or subtracting like terms, add or subtract only their coefficients.

Since \(63=9\times7\) and \(28=4\times7\), \(\sqrt{63}=3\sqrt{7}\) and \(\sqrt{28}=2\sqrt{7}\). Therefore, \(\sqrt{63}-\sqrt{28}=3\sqrt{7}-2\sqrt{7}=\sqrt{7}\). \(2\sqrt{7}\) is only the simplified form of \(\sqrt{28}\), not the difference. Exam tip: simplify surds into like terms before subtracting them.

Given \(x=\sqrt{15}\), we get \(x^2=(\sqrt{15})^2=15\). Therefore, \(x^2+10=15+10=25\), so option C is correct. Option 20 would result from adding 5 instead of 10. Exam tip: squaring a square root gives the original non-negative number.

\(\sqrt{49}=7\), and 7 is a real number, so option C is correct. Options A, B, and D contain square roots of negative numbers, which are not defined in the real number system. Exam tip: A square root is real only when the number inside the radical is zero or positive.

Since \(300=100\times3=10^2\times3\), \(\sqrt{300}=\sqrt{10^2\times3}=10\sqrt{3}\). The close distractor \(5\sqrt{3}\) squares to \(75\), not \(300\). Exam tip: to simplify a surd, first identify the greatest perfect-square factor of the number.

\(\sqrt{8}=\sqrt{4\times2}=2\sqrt{2}\). Therefore, \(a=\sqrt{2}+2\sqrt{2}=3\sqrt{2}\), so option B is correct. \(2\sqrt{2}\) is only the simplified form of \(\sqrt{8}\); the remaining \(\sqrt{2}\) must still be added. Exam tip: simplify surds by identifying perfect-square factors inside the radical.

Rational numbers can be written as fractions, integers, terminating decimals or repeating decimals. In option A, \(0.125\) is a terminating decimal (\(\tfrac{1}{8}\)), \(-4\) is an integer, and \(\sqrt{169}=13\) is an integer — so all are rational. A common distractor is D: \(\sqrt{18}=3\sqrt{2}\) is irrational, so D is not all rational. Exam tip: check whether a square root is of a perfect square and whether decimals terminate or repeat to decide rationality quickly.

\(175=25\times7=5^2\times7\). Therefore, \(\sqrt{175}=\sqrt{5^2\times7}=5\sqrt{7}\). \(7\sqrt{5}\) is not correct because its square is \(245\), not \(175\). In exams, identify the greatest perfect-square factor before simplifying a surd.

Given \(\sqrt{x}=15\), square both sides to get \(x=15^2=225\). Therefore, 225 is correct. The value 30 is obtained by doubling 15, but removing a square root requires squaring both sides, not doubling. Exam tip: if \(\sqrt{x}=a\), write \(x=a^2\).

Since \(242=121\times 2=11^2\times 2\), \(\sqrt{242}=\sqrt{11^2\times 2}=11\sqrt{2}\). Hence, \(11\sqrt{2}\) is correct. Squaring \(22\sqrt{2}\) gives \(968\), not \(242\). Exam tip: To simplify a surd, first identify the greatest perfect-square factor of the number.

\(\sqrt{100}=10\), and 10 is a real number. The square root of a negative number is not defined within the real number system; therefore, \(\sqrt{-36}\), \(\sqrt{-49}\), and \(\sqrt{-64}\) are not real numbers. Exam tip: for a square root to be real, the number inside the root must be zero or positive.

Since \(48=16\times3\), \(\sqrt{48}=\sqrt{16\times3}=4\sqrt{3}\). Likewise, \(75=25\times3\), so \(\sqrt{75}=5\sqrt{3}\). Hence, \(4\sqrt{3}+5\sqrt{3}=9\sqrt{3}\), making option B correct. Writing \(\sqrt{48}+\sqrt{75}\) as \(\sqrt{123}\) is incorrect because square roots cannot be added in that way. Exam tip: factor each radicand using its greatest perfect-square factor before simplifying.

Here, \(x=\sqrt{7}+\sqrt{5}\). Therefore, \(x^2=(\sqrt{7}+\sqrt{5})^2=7+5+2\sqrt{7}\sqrt{5}=12+2\sqrt{35}\). Hence, option A is correct. In option B, the middle term \(2ab\) has been taken incompletely. Exam tip: while squaring a binomial, always include the middle term \(2ab\).

Using the quotient rule for square roots, \(\frac{\sqrt{180}}{\sqrt{5}}=\sqrt{\frac{180}{5}}=\sqrt{36}=6\). Therefore, 6 is correct. Dividing 180 by 5 gives 36, not 25, so option 5 is not correct. Exam tip: use \(\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\) when \(b\) is positive.

Write \(18=9\times2\) and \(8=4\times2\). Then \(\sqrt{18}=3\sqrt{2}\) and \(\sqrt{8}=2\sqrt{2}\). Hence, \(a=3\sqrt{2}-2\sqrt{2}=\sqrt{2}\). \(2\sqrt{2}\) is a close distractor because it is only the simplified form of \(\sqrt{8}\), not the difference of the two surds. Exam tip: first extract perfect-square factors from radicals, then combine like surds.

Since \(49=7^2\) and \(81=9^2\), \(\sqrt{49}=7\) and \(\sqrt{81}=9\). Therefore, \(\sqrt{49}+\sqrt{81}=7+9=16\). Option 15 is incorrect because the square roots must first be evaluated and then added. Exam tip: recognise square roots of perfect squares directly, such as \(49\to7\) and \(81\to9\).

\(\sqrt{64}=8\) and \(\sqrt{16}=4\). Therefore, \(\sqrt{64}-\sqrt{16}=8-4=4\). Option 5 would result from an incorrect subtraction. Exam tip: evaluate the square roots of perfect squares separately before performing the operation.

Since \(121=11^2\), \(\sqrt{121}=11\). Therefore, \(\sqrt{121}\div11=11\div11=1\). Option 11 is only the value of the square root; the division by 11 still has to be performed. Exam tip: evaluate the square root first, then carry out the indicated division or multiplication.

Write \(8=4\times2\). Then \(\sqrt{8}=\sqrt{4\times2}=\sqrt{4}\times\sqrt{2}=2\sqrt{2}\). Hence, the correct simplified form is \(2\sqrt{2}\). The expression \(4\sqrt{2}\) is incorrect because its square is 32, not 8. Exam tip: To simplify a surd, first separate the greatest perfect-square factor inside the radical.

\(\sqrt{3}\approx 1.732\), whereas \(\frac{9}{5}=1.8\), \(2=2\), and \(\sqrt{5}\approx 2.236\). Hence, \(\sqrt{3}\) is the smallest. \(\frac{9}{5}\) is the closest distractor, but it is greater because it equals \(1.8\). Exam tip: compare square roots using nearby perfect squares or decimal approximations.

\(\sqrt{9}=3\) and \(\sqrt{16}=4\), so \(x=3+4=7\). Option 6 is incorrect because it is not the sum of the two square roots. Exam tip: evaluate each perfect-square root first, then add or subtract.

The set of real numbers consists of all rational as well as all irrational numbers. For example, \(\frac{3}{4}\) is rational and \(\sqrt{2}\) is irrational, but both are real numbers. The set of rational numbers, option D, is the closest distractor but it does not include irrational numbers. Exam tip: remember that \(\mathbb{R}=\mathbb{Q}\cup\) irrational numbers.

\(\sqrt{144}=12\) because \(12^2=144\), and \(\sqrt{1}=1\) because \(1^2=1\). Therefore, \(\sqrt{144}+\sqrt{1}=12+1=13\). Option 12 gives only \(\sqrt{144}\) and misses adding \(\sqrt{1}\). Exam tip: evaluate each square root separately before performing the operation.

Since \(13 \times 13 = 169\), \(\sqrt{169}=13\). The square of 12 is 144, so 12 is not correct. Exam tip: verify a square root by squaring the option and matching it with the given number.

\(\sqrt{196}=14\) and \(\sqrt{49}=7\). Therefore, \(\sqrt{196}\div\sqrt{49}=14\div7=2\). Hence, 2 is the correct option. Option 1 may result from an error in division. Exam tip: evaluate each perfect square root first, then perform the remaining operation.