Real Numbers

( \sqrt{2}\times\sqrt{8}=\sqrt{16}=4 ). In multiplication, numbers inside square roots can be multiplied.

Using the quotient rule for square roots, \(\frac{\sqrt{27}}{\sqrt{3}}=\sqrt{\frac{27}{3}}=\sqrt{9}=3\). Therefore, option A is correct. Option C results from forgetting to take the square root of 9, while option D is only the simplified form of \(\sqrt{27}\) and does not account for division by \(\sqrt{3}\). Exam tip: when the radicands are positive, combine the quotient under one square root before simplifying.

Since \(2^2=4\) and \(3^2=9\), and \(4<6<9\), it follows that \(2<\sqrt{6}<3\). Therefore, \(\sqrt{6}\) is correct. Exam tip: for positive numbers, compare the numbers under the square roots with the squares of the given bounds.

( \sqrt{49}=7 ), and (7) is a rational real number. The square root of a perfect square can be an integer.

( \pi ) is an irrational number and lies on the real number line. Do not treat it as exactly equal to ( \frac{22}{7} ).

( \frac{22}{7} ) is a ratio of two integers with non-zero denominator. So it is a rational real number.

( \frac{1}{2} ), ( \sqrt{3} ), and ( -5 ) are all real numbers. A fraction with zero denominator is not real.

The governing concept is the principal square root and the effect of an outside negative sign. The symbol √4 denotes the non-negative principal square root of 4, so √4 = 2. The minus sign is written before the radical and therefore changes the result after the square root is evaluated: −√4 = −(2) = −2. Thus option B is correct. Option A would be correct for √4 without the outside negative sign. Option C is the radicand, not its square root, while option D incorrectly treats the square root of 4 as 4 and also applies a negative sign. It is important not to confuse −√4 with √(−4), which is not a real number.

Absolute value represents a number’s distance from zero, so it is never negative. The distance of −7 from zero is 7; therefore, \(|-7|=7\). Option A is incorrect because it gives the signed value, not the absolute value. Exam tip: To find the absolute value of a negative number, remove its minus sign.

There are infinitely many real numbers between any two distinct real numbers. This is an important property of the number line.

\(\sqrt{36}=6\) and \(\sqrt{64}=8\), so \(6<8\), which gives \(\sqrt{36}<\sqrt{64}\). Therefore, option C is correct. Option A reverses the inequality, while option B incorrectly treats the two values as equal. Option D is also incorrect because 6 and 8 are both real and rational numbers. Exam tip: For perfect squares, calculate their square roots first and then compare the results.

The governing concept is the identity property of addition. An additive identity is a number that leaves every number unchanged when it is added to that number. For any real number x, the defining relation is x + 0 = 0 + x = x. Therefore, zero is the additive identity in the set of real numbers, and option C is correct. Option A is not an additive identity because x + 1 is generally greater than x. Similarly, adding −1 changes most numbers, so option B cannot be correct. Adding 2 also changes the original number. Notice that 1 is the multiplicative identity because x × 1 = x; this distinction helps avoid confusing the identities for addition and multiplication.

Multiplying any real number by (1) gives the same number. Hence (1) is the multiplicative identity.

The additive inverse of a number is the number that gives a sum of 0 when added to the original number. Here, \((-9)+9=0\), so the additive inverse of \((-9)\) is 9. \(\frac{1}{9}\) is its multiplicative inverse, not its additive inverse. Exam tip: to find an additive inverse, change the sign of the number.

The multiplicative inverse of a non-zero number is the number that gives product 1 when multiplied by the original number. For a non-zero fraction \\(\frac{a}{b}\\), the inverse is \\(\frac{b}{a}\\), because \\(\frac{a}{b}\times\frac{b}{a}=1\\). Here the numerator 4 and denominator 7 are interchanged, so the inverse of \\(\frac{4}{7}\\) is \\(\frac{7}{4}\\). Therefore, option A is correct.

We can verify the result directly: \\(\frac{4}{7}\times\frac{7}{4}=\frac{28}{28}=1\\). The negative choices are not correct because multiplying \\(\frac{4}{7}\\) by a negative version gives a negative product, not 1. The number \\(\frac{11}{7}\\) also does not produce 1. The fraction is non-zero, so its reciprocal exists.

Zero is neither positive nor negative. Positive numbers are greater than 0, such as 1 and \(\frac{1}{2}\), whereas negative numbers are less than 0, such as -1. Therefore, 0 is the correct answer. Exam tip: On the number line, numbers to the right of 0 are positive and those to the left are negative.

\(\sqrt{121}\) is the positive number whose square is 121. Since \(11^2=121\), we get \(\sqrt{121}=11\). Option 121 is the radicand, not its square root. Exam tip: for a positive number \(a\), \(\sqrt{a^2}=a\).

The governing concept is extracting a perfect-square factor from a radical. Factor 12 as 4 × 3, where 4 is a perfect square. Using √(ab) = √a × √b for non-negative factors, √12 = √(4 × 3) = √4 × √3 = 2√3. Therefore option B is correct. Option A, 6√2, would square to 72 and is too large. Option C, 3√2, would square to 18, not 12. Option D, 4√3, would square to 48. The expression 2√3 cannot be reduced further because 3 has no factor greater than 1 that is a perfect square. This method is preferable to using a decimal approximation because it gives the exact simplified radical.

For positive numbers, the square root of the greater number is also greater. Since (6>5), ( \sqrt{6}>\sqrt{5} ).

Adding the irrational ( \sqrt{2} ) to the rational number (3) generally gives an irrational number. It is also real.