Real Numbers

Here, \(x=\sqrt{2}+\sqrt{3}\). Therefore, \(x^2=(\sqrt{2}+\sqrt{3})^2=2+3+2\sqrt{2}\sqrt{3}=5+2\sqrt{6}\). Hence, option A is correct. In option B, the coefficient 2 in the middle term \(2ab\) has been omitted. Exam tip: When squaring two surds, always include the middle term \(2ab\).

Since \(72=36\times2\), and \(36\) is a perfect square, \(\sqrt{72}=\sqrt{36\times2}=6\sqrt{2}\). Therefore, \(6\sqrt{2}\) is correct. For example, \(3\sqrt{2}\) squares to \(18\), not \(72\). Exam tip: to simplify a surd, first identify the greatest perfect-square factor of the number.

Given \(a=\sqrt{5}-2\), adding 2 gives \(a+2=\sqrt{5}\). Therefore, \((a+2)^2=(\sqrt{5})^2=5\). Option 10 would result from incorrectly evaluating the square of \(\sqrt{5}\). Exam tip: use the identity \((\sqrt{x})^2=x\).

Since \(18=9\times2\), \(\sqrt{18}=3\sqrt{2}\); and since \(50=25\times2\), \(\sqrt{50}=5\sqrt{2}\). Therefore, \(\sqrt{18}+\sqrt{50}=3\sqrt{2}+5\sqrt{2}=8\sqrt{2}\). The option \(7\sqrt{2}\) would result from adding the coefficients incorrectly. Exam tip: first take out perfect-square factors from surds, then combine like surds.

\(\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}\). Therefore, \(a+b=\sqrt{3}+2\sqrt{3}=3\sqrt{3}\). \(2\sqrt{3}\) is only the simplified value of \(b\), not the sum. Exam tip: simplify surds to like radicals before adding them.

Given \(\sqrt{x}=9\), squaring both sides gives \(x=9^2=81\). Hence, 81 is correct. The number 9 is the value of \(\sqrt{x}\), not of \(x\). Exam tip: After squaring a square-root equation, check the value in the original equation.

Since \(98=49\times2\) and \(8=4\times2\), \(\sqrt{98}=7\sqrt{2}\) and \(\sqrt{8}=2\sqrt{2}\). Therefore, \(\sqrt{98}-\sqrt{8}=7\sqrt{2}-2\sqrt{2}=5\sqrt{2}\), so option C is correct. \(4\sqrt{2}\) would result only if the coefficients differed by 4. Exam tip: first take perfect-square factors out of each surd, then subtract like surd terms.

Given \(x=\sqrt{2}-1\), we get \(x(\sqrt{2}+1)=(\sqrt{2}-1)(\sqrt{2}+1)\). This matches \((a-b)(a+b)=a^2-b^2\), so its value is \((\sqrt{2})^2-1^2=2-1=1\). Option 2 would result from forgetting to subtract \(1^2\). Exam tip: identify conjugate factors and use the difference-of-squares identity.

Since \(45=9\times5\) and \(80=16\times5\), \(\sqrt{45}=3\sqrt{5}\) and \(\sqrt{80}=4\sqrt{5}\). Therefore, \(\sqrt{45}+\sqrt{80}=3\sqrt{5}+4\sqrt{5}=7\sqrt{5}\). A result such as \(9\sqrt{5}\) can arise from simplifying the square roots incorrectly. Exam tip: first factor out perfect squares, such as \(9\) or \(16\), from under each radical.

\(\sqrt{2}\approx 1.414\), \(\frac{3}{2}=1.5\), and \(\sqrt{3}\approx 1.732\). All three are greater than 1, so 1 is the smallest number. \(\sqrt{2}\) may seem like the closest distractor, but it is still greater than 1. Exam tip: use approximate decimal values to compare square roots quickly.

Given \(a=\sqrt{8}\), we get \(a^2=(\sqrt{8})^2=8\). Squaring the principal square root of a positive number gives back the number itself. Option 4 is not the square of \(\sqrt{8}\); it is the value of \(\sqrt{16}\). Exam tip: use \((\sqrt{x})^2=x\) and write the radicand directly.

Since \(200=100\times2\) and \(100\) is a perfect square, \(\sqrt{200}=\sqrt{100}\times\sqrt{2}=10\sqrt{2}\). The option \(5\sqrt{2}\) squares to \(50\), so it is not equal to \(\sqrt{200}\). Exam tip: To simplify a surd, first identify the greatest perfect-square factor of the number.

Both terms are like surds: \(\sqrt{5}\) and \(\sqrt{5}\). Hence, add their coefficients: \(1\sqrt{5}+1\sqrt{5}=2\sqrt{5}\). Therefore, the correct answer is \(2\sqrt{5}\). \(\sqrt{10}\) is incorrect because \(\sqrt{5}+\sqrt{5}\) cannot be written as \(\sqrt{5+5}\). Exam tip: add or subtract only the coefficients of like surds.

For positive numbers, \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\). Therefore, \(\sqrt{27}\times\sqrt{3}=\sqrt{81}=9\). The number \(27\) is not the square root of \(81\). Exam tip: Combine the radicals first and then look for a perfect square.

\(b=\sqrt{24}=\sqrt{4\times6}=2\sqrt{6}\), while \(a=\sqrt{6}\). Therefore, \(b-a=2\sqrt{6}-\sqrt{6}=\sqrt{6}\). \(2\sqrt{6}\) is the value of \(b\), not the difference. Exam tip: simplify surds to the same radicand before subtracting them.

The set of real numbers is the union of rational and irrational numbers: \(\mathbb{R}=\mathbb{Q}\cup\mathbb{I}\). Rational numbers can be written in the form \(p/q\), whereas irrational numbers cannot be written in this form. Integers, natural numbers, and whole numbers are all subsets of rational numbers, so they do not alone cover all real numbers. Exam tip: remember irrational examples such as \(\sqrt{2}\) and \(\pi\).

For positive numbers, \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\). Hence, \(\sqrt{12}\times\sqrt{75}=\sqrt{12\times75}=\sqrt{900}=30\). Option 15 is incorrect because \(\sqrt{900}=30\), not 15. Exam tip: Combine surds under one square root and look for a perfect square.

Given \(x=\sqrt{3}+\sqrt{2}\), \(x^2=(\sqrt{3}+\sqrt{2})^2=3+2+2\sqrt{3}\sqrt{2}=5+2\sqrt{6}\). Therefore, \(x^2-5=2\sqrt{6}\). The option \(\sqrt{6}\) misses the factor 2 in the middle term. Exam tip: while using \((a+b)^2=a^2+b^2+2ab\), do not omit \(2ab\).

Option A simplifies to \((-7/5)/(-7/5)=1\). The number 1 is an integer and can be written as a ratio of integers (\(p/q\), \(q\neq0\)), so it is rational. Option B, \(\sqrt{21}\), is not a perfect square and therefore irrational. Option C is \(\sqrt{31}/5\); dividing a nonzero irrational by a nonzero rational gives an irrational number, so C is irrational. Option D contains \(\pi\), which is irrational, so \((\pi+1)/2\) remains irrational. Exam tip: simplify expressions first — identical nonzero numerator and denominator give 1 (rational); check for perfect squares and known irrational constants to identify irrational options quickly.

Since \(147=49\times 3=7^2\times 3\), \(\sqrt{147}=\sqrt{7^2\times3}=7\sqrt{3}\). The option \(3\sqrt{7}\) would correspond to \(\sqrt{9\times7}=\sqrt{63}\), so it is not correct. Exam tip: identify the greatest perfect-square factor before simplifying a surd.

Given \(\sqrt{x}=12\), squaring both sides gives \(x=12^2=144\). Therefore, \(x-44=144-44=100\). The value 144 is the value of \(x\), not of the required expression \(x-44\). Exam tip: First square a square-root equation to find the variable, then substitute it into the expression asked.

Irrational numbers have non-terminating, non-repeating decimal expansions. The square root of a perfect square is rational (for example \(\sqrt{4}=2\)), while the square root of a non-perfect square is irrational. In option A, \(\sqrt{6},\ \sqrt{10},\ \sqrt{15}\) are not square roots of perfect squares, so all three are irrational. The closest distractor B is wrong because \(\sqrt{4}=2\) is rational. Options C and D are also incorrect since \(\tfrac{1}{2}\) and \(0\) are rational. Exam tip: when checking square roots, first test if the radicand is a perfect square — if it is, the root is rational.

Since \(32=16\times2\), \(\sqrt{32}=\sqrt{16\times2}=4\sqrt{2}\). Similarly, \(8=4\times2\), so \(\sqrt{8}=2\sqrt{2}\). Therefore, \(4\sqrt{2}+2\sqrt{2}=6\sqrt{2}\), making option B correct. \(4\sqrt{2}\) is only the value of \(\sqrt{32}\), not their sum. Exam tip: simplify surds into like radical terms, such as \(\sqrt{2}\), before adding them.

Given \(a=\sqrt{10}\), squaring gives \(a^2=(\sqrt{10})^2=10\). Therefore, \(\frac{a^2}{2}=\frac{10}{2}=5\). The value \(10\) is only \(a^2\); it still needs to be divided by 2. Exam tip: remember that \((\sqrt{x})^2=x\).

Since \(108=36\times3\), and \(36\) is a perfect square, \(\sqrt{108}=\sqrt{36\times3}=\sqrt{36}\sqrt{3}=6\sqrt{3}\). The option \(3\sqrt{6}\) is not correct because its square is \(54\), not \(108\). Exam tip: To simplify a surd, first identify the greatest perfect-square factor of the number.