Real Numbers

\(\sqrt{8}=\sqrt{4\times2}=2\sqrt{2}\). Therefore, \(a+b=\sqrt{2}+2\sqrt{2}=3\sqrt{2}\), so option B is correct. \(2\sqrt{2}\) is only the simplified value of \(b\), not the sum. Exam tip: simplify surds first, then add like surd terms.

Since \(45=9\times5\), \(\sqrt{45}=\sqrt{9\times5}=3\sqrt{5}\). Therefore, \(\sqrt{45}-\sqrt{5}=3\sqrt{5}-\sqrt{5}=2\sqrt{5}\). \(3\sqrt{5}\) is only the simplified form of \(\sqrt{45}\), not the result after subtraction. Exam tip: before adding or subtracting surds, first extract perfect-square factors from the radicands.

Statement C is false because the set of real numbers includes both rational and irrational numbers. For example, \(\sqrt{2}\) is a real number, but it cannot be written in the form \(p/q\); hence, it is not rational. Statement D is correct because real numbers are the union of rational and irrational numbers. Exam tip: Examples such as \(\sqrt{2}\) and \(\pi\) help identify irrational real numbers.

Using the quotient rule for square roots, \(\frac{\sqrt{32}}{\sqrt{2}}=\sqrt{\frac{32}{2}}=\sqrt{16}=4\). Therefore, 4 is correct. The number 16 is only the value inside the square root; its square root must still be evaluated. Exam tip: use \(\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\) when \(b>0\).

The blocks of 0s keep increasing in length, so no fixed block of digits repeats. A non-terminating, non-repeating decimal is irrational. Exam tip: check repetition, not merely the digits used.

Both terms are like surds: \(\sqrt{7}+\sqrt{7}=1\sqrt{7}+1\sqrt{7}=(1+1)\sqrt{7}=2\sqrt{7}\). \(\sqrt{14}\) is incorrect because \(\sqrt{a}+\sqrt{b}\) cannot generally be written as \(\sqrt{a+b}\). Exam tip: add only the coefficients of like surds.

(-4) is a negative integer, so it belongs to the set of integers. Natural numbers and whole numbers do not include negative numbers. Although (-4) is also a rational number, integers form the smaller set containing it. Exam tip: Remember the usual classification order: natural numbers, whole numbers, integers, rational numbers, and real numbers.

Since \(48=16\times3\), \(\sqrt{48}=\sqrt{16\times3}=4\sqrt{3}\). Therefore, \(\sqrt{48}-2\sqrt{3}=4\sqrt{3}-2\sqrt{3}=2\sqrt{3}\), so option A is correct. \(4\sqrt{3}\) is only the simplified form of \(\sqrt{48}\); the subtraction of \(2\sqrt{3}\) must still be done. Exam tip: first identify the largest perfect-square factor inside a surd.

\(\sqrt{81}=9\) because \(9^2=81\), and \(\sqrt{1}=1\) because \(1^2=1\). Therefore, \(\sqrt{81}+\sqrt{1}=9+1=10\). Option 9 is only the value of \(\sqrt{81}\); it misses the addition of \(\sqrt{1}\). Exam tip: the square root of a perfect square is the positive number whose square equals the given number.

Since \(20=4\times5\), \(\sqrt{20}=\sqrt{4\times5}=2\sqrt{5}\). Therefore, \(\sqrt{20}+\sqrt{5}=2\sqrt{5}+\sqrt{5}=3\sqrt{5}\), so option A is correct. \(2\sqrt{5}\) is only the simplified form of \(\sqrt{20}\), not of the complete sum. Exam tip: simplify surds first, then combine like surds.

\(\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}\) and \(\sqrt{8}=\sqrt{4\times2}=2\sqrt{2}\). Therefore, \(\sqrt{50}-\sqrt{8}=5\sqrt{2}-2\sqrt{2}=3\sqrt{2}\), so option C is correct. \(\sqrt{42}\) is incorrect because the difference of square roots is not generally equal to \(\sqrt{50-8}\). Exam tip: simplify each surd by factoring out its greatest perfect-square factor first.

\(0.\overline{27}=0.272727\ldots\) is a non-terminating recurring decimal, and it equals \(\frac{27}{99}=\frac{3}{11}\). Hence, it is rational even though its decimal expansion does not end. \(\sqrt{2}\), \(\pi\), and \(0.101001000100001\ldots\) have non-terminating, non-recurring decimal expansions, so they are irrational. Exam tip: A non-terminating recurring decimal is rational; a non-terminating non-recurring decimal is irrational.

Here, \(x\times x=(\sqrt{6})\times(\sqrt{6})=(\sqrt{6})^2=6\). Squaring the principal square root of a positive number gives the number itself. \(\sqrt{12}\) is incorrect because \(\sqrt{6}\times\sqrt{6}=\sqrt{36}=6\). Exam tip: recognise directly that \(\sqrt{a}\times\sqrt{a}=a\).

\(0.27272727\ldots\) is a non-terminating recurring decimal because the block 27 repeats indefinitely. Every recurring decimal is rational; in fact, \(0.272727\ldots=\frac{3}{11}\). Hence, it disproves the student's claim. In option B, the number of zeros keeps increasing, so there is no fixed repeating block and it is irrational. Exam tip: For a non-terminating decimal, check whether a repeating pattern exists before deciding if it is rational or irrational.

Given \(a=\sqrt{3}\), we get \(2a+\sqrt{3}=2\sqrt{3}+\sqrt{3}=3\sqrt{3}\). Hence, \(3\sqrt{3}\) is correct. \(2\sqrt{3}\) is only the value of \(2a\); the remaining \(\sqrt{3}\) must still be added. Exam tip: for like surds, add their coefficients.

Irrational numbers are not rational numbers, so statement C is incorrect. A rational number can be written in the form \(p/q\), where \(p\) and \(q\) are integers and \(q\ne0\). Integers, natural numbers, and whole numbers are all included in the set of real numbers. Exam tip: Remember the inclusion order: natural ⊂ whole ⊂ integers ⊂ rational ⊂ real.

Since \(24=4\times6\) and \(54=9\times6\), \(\sqrt{24}=2\sqrt{6}\) and \(\sqrt{54}=3\sqrt{6}\). Therefore, \(\sqrt{24}+\sqrt{54}=2\sqrt{6}+3\sqrt{6}=5\sqrt{6}\). A result such as \(7\sqrt{6}\) can arise from adding the coefficients incorrectly. Exam tip: first take out perfect-square factors from each surd, then add the coefficients of like surds.

\(\frac{9}{2}=4.5\), which is not an integer because integers do not have a fractional or decimal part. In contrast, \(-8\), \(0\), and \(\sqrt{121}=11\) are all integers. Exam tip: simplify a square-root option first; if its value is a whole number, it is an integer.

A rational number has a decimal expansion that either terminates or repeats after some point. Here, the number of zeros between the 1s keeps increasing, so no fixed block of digits repeats periodically. Hence \(0.101001000100001\ldots\) is non-terminating and non-repeating, making it irrational. Using only the digits 0 and 1 does not make a number rational. Exam tip: When a decimal has a pattern, check whether a fixed block actually repeats after a certain point.

Since \(72=36\times2\) and \(36\) is a perfect square, \(\sqrt{72}=\sqrt{36\times2}=\sqrt{36}\sqrt{2}=6\sqrt{2}\). \(4\sqrt{2}\) is incorrect because its square is \(32\), not \(72\). Exam tip: To simplify a surd, first identify the greatest perfect-square factor of the number.