Real Numbers

(5^2=25) and (6^2=36), so ( \sqrt{26} ) lies between (5) and (6). Compare nearby perfect squares.

( \sqrt{20}\approx4.47 ), so ( -\sqrt{20}\approx-4.47 ). ( -4.4 ) is closer to zero, so it is greater.

The governing concepts are the classification of real numbers and the fact that division by zero is undefined. Every rational and every irrational number belongs to the real-number system. Thus √2 is real even though it is irrational. The repeating decimal 0.333… equals 1/3, so it is rational and real. The integer −7 can be written as −7/1, making it rational and therefore real. However, 5/0 is not defined: if 5/0 were a number x, then multiplying by zero would require 0×x = 5, which is impossible because 0×x is always 0. Hence 5/0 is not a real number, and option C is the incorrect statement.

The diagonal of a square is side × \(\sqrt{2}\), so it is \(\sqrt{2}\times\sqrt{2}=2\) cm, which is rational. The perimeter is \(4\times\sqrt{2}=4\sqrt{2}\) cm, which is irrational because the product of a non-zero rational number and an irrational number is irrational. Therefore, option A is correct. Exam tip: Do not classify an expression merely by looking at an irrational term; simplify it first.

A rational number has a terminating decimal expansion when, after simplification, the denominator has no prime factors other than 2 and 5. This happens because powers of 2 and 5 can combine to make a power of 10. For \\(\frac{13}{40}\\), the numerator and denominator have no common factor, and \\(40=2^3\times5\\).

Since the denominator contains only the allowed prime factors 2 and 5, the decimal terminates. In fact, \\(\frac{13}{40}=\frac{13\times25}{40\times25}=\frac{325}{1000}=0.325\\). Therefore, option A is correct. It is not irrational or undefined, and it is not non-terminating non-repeating; those descriptions do not fit this rational fraction.

Factoring 18 gives \(18=9\times2\), so \(\sqrt{18}=3\sqrt{2}\). Since \(\sqrt{2}\) is irrational, multiplying it by the non-zero rational number 3 still gives an irrational number. Therefore, \(\sqrt{18}\) is irrational. Option A is wrong because the square root of every integer is not rational, and 18 is not a perfect square. Exam tip: Before classifying \(\sqrt{n}\), check whether \(n\) is a perfect square.

\(\sqrt{5}\times\sqrt{20}=\sqrt{5\times20}=\sqrt{100}=10\). The number 10 is an integer and a natural number, so it is also a rational real number. Therefore, option A is correct. It is not irrational because the product simplifies to a whole number. Exam tip: For positive radicands, use \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\), then check whether the resulting square root is a perfect square.

The direct answer is B, \\(-9\\). The two expressions are conjugates: \\(2+\\sqrt{13}\\) and \\(2-\\sqrt{13}\\). Use the difference-of-squares identity \\((a+b)(a-b)=a^2-b^2\\). Here, \\(a=2\\) and \\(b=\\sqrt{13}\\). Therefore, \\((2+\\sqrt{13})(2-\\sqrt{13})=2^2-(\\sqrt{13})^2=4-13=-9\\). Option A, 9, has the wrong sign because 4 is smaller than 13. Option B, -9, is correct. Option C, 17, would result from adding 4 and 13, but multiplication of conjugates requires subtraction. Option D, \\(4+\\sqrt{13}\\), is not the result of multiplying the two binomials and leaves an unnecessary radical. You can also check by expansion: \\(4-2\\sqrt{13}+2\\sqrt{13}-13=4-13=-9\\); the middle terms cancel. Memory cue: conjugates multiply to the square of the rational part minus the square of the radical part. The supplied answer is correct.

Multiplying by the conjugate ( \sqrt{5}-1 ) makes the denominator (5-1=4). Use difference of squares in the conjugate method.

A rational number has a terminating or recurring decimal expansion. Here the gaps of zeros between 1s are 1, 2, 3, 4, ..., so no fixed repeating block exists. Exam tip: check the repetition pattern, not merely the digits used.

In 0.1010010001..., the number of zeros between successive 1s is 1, 2, 3, 4..., so no repeating block occurs. A non-terminating, non-repeating decimal is irrational. Exam tip: an infinite decimal can still be rational only if it repeats.

In real numbers, ( \sqrt{a^2}=|a| ) because the principal square root is non-negative. Pay attention to the sign.

Using the identity \(\sqrt{x^2}=|x|\), we get \(\sqrt{(-5)^2}=|-5|=5\). The principal square root is always non-negative, so \(-5\) is not correct. Exam tip: remember that \(\sqrt{x^2}=|x|\), not always \(x\).

Multiplying by the conjugate ( \sqrt{3}-\sqrt{2} ) makes the denominator (3-2=1). So the rationalised form is ( \sqrt{3}-\sqrt{2} ).

Since \(180=36\times5\) and \(36\) is the largest perfect-square factor, \(\sqrt{180}=\sqrt{36\times5}=\sqrt{36}\times\sqrt{5}=6\sqrt{5}\). Option B is the unsimplified form, while options C and D have incorrect coefficients. Exam tip: factor the radicand using its largest perfect-square factor before simplifying the surd.

\(245=49\times5=7^2\times5\). Therefore, \(\sqrt{245}=\sqrt{7^2\times5}=7\sqrt{5}\), so option B is correct. Option D is incorrect because the square root of the perfect-square factor \(49\) is \(7\), not \(49\). Exam tip: To simplify a radical, factor the radicand into the largest possible perfect square times the remaining factor.

\(288=144\times2=12^2\times2\). Therefore, \(\sqrt{288}=\sqrt{12^2\times2}=12\sqrt{2}\), so option C is correct. In option A, the coefficient is doubled, while options B and D are not equal to \(\sqrt{288}\). Exam tip: factor out the largest perfect-square factor before simplifying a square root.

\(\sqrt{45}=\sqrt{9\times5}=3\sqrt{5}\), so \(2\sqrt{45}=6\sqrt{5}\). Similarly, \(\sqrt{20}=\sqrt{4\times5}=2\sqrt{5}\), so \(3\sqrt{20}=6\sqrt{5}\). Adding the like surd terms gives \(6\sqrt{5}+6\sqrt{5}=12\sqrt{5}\), so option A is correct. An answer such as option B results from adding the coefficients incorrectly. Exam tip: simplify each surd first, then add only terms with the same surd part.

\(\sqrt{27}=3\sqrt{3}\) and \(\sqrt{75}=5\sqrt{3}\). Therefore, \(4\sqrt{27}-2\sqrt{75}=4(3\sqrt{3})-2(5\sqrt{3})=12\sqrt{3}-10\sqrt{3}=2\sqrt{3}\). Hence, option A is correct. Exam tip: Simplify each surd first, then combine only like surd terms.

By distributive law, ( \sqrt{5}\times3+\sqrt{5}\times\sqrt{5}=3\sqrt{5}+5 ). Be careful in surd multiplication.

Apply the identity \((a+b)^2=a^2+2ab+b^2\). Here, \(a=4\) and \(b=\sqrt{3}\), so \((4+\sqrt{3})^2=4^2+2(4)(\sqrt{3})+(\sqrt{3})^2=16+8\sqrt{3}+3=19+8\sqrt{3}\). Therefore, option A is correct. Remember that \((\sqrt{3})^2=3\), not \(\sqrt{4}\).

The direct answer is option B: \(44-24\sqrt{2}\). Start with \((a-b)^2=a^2-2ab+b^2\). Here \(a=6\) and \(b=\sqrt{8}\), so \((6-\sqrt{8})^2=36-12\sqrt{8}+8=44-12\sqrt{8}\). Since \(\sqrt{8}=2\sqrt{2}\), this becomes \(44-24\sqrt{2}\). Option A keeps \(\sqrt{8}\) and has the wrong middle term. Option B is correct after simplification. Option C has an incorrect constant term and radical coefficient. Option D incorrectly treats the cross product as involving \(\sqrt{6}\). Remember: square the first term, subtract twice the product, then square the second term.

This is a product of conjugate expressions. Using \((a+b)(a-b)=a^2-b^2\), we get \((5+\sqrt{17})(5-\sqrt{17})=5^2-(\sqrt{17})^2=25-17=8\). Therefore, the correct answer is 8. The value 42 results from adding 25 and 17, but this product requires their difference. Exam tip: Whenever you see \((a+b)(a-b)\), apply \(a^2-b^2\) directly.

Since \\(50=25\times2\\), we get \\(\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}\\). Because \\(\sqrt{2}\\) is irrational, multiplying it by the non-zero rational number 5 still gives an irrational number. Therefore, option A is correct. Option C has the correct simplification but incorrectly classifies the result as rational. Exam tip: take perfect-square factors outside the square root and check whether a non-perfect-square factor remains inside.

Multiplying numerator and denominator by ( \sqrt{2} ) gives ( \frac{5\sqrt{2}}{6} ). The radical should be removed from the denominator.