Irrational numbers

Step 1: The two terms are opposites of each other. Step 2: (\sqrt{13}+(-\sqrt{13})=0). Step 3: The sum of opposite terms is always zero.

Step 1: Dividing by a non-zero rational number does not remove irrationality. Step 2: For example, (\frac{\sqrt{2}}{5}) is irrational. Step 3: The condition (r\neq0) is necessary because division by zero is not possible.

Step 1: (275=25 \times 11). Step 2: (\sqrt{275}=\sqrt{25 \times 11}=5\sqrt{11}). Step 3: Take the perfect square factor outside to simplify the answer.

Step 1: Recurring or terminating decimals are rational. Step 2: A non-terminating decimal with no fixed repetition can be irrational. Step 3: If no repeating rule is visible, examine the number carefully.

Step 1: Write (243=81 \times 3). Step 2: (\sqrt{243}=\sqrt{81 \times 3}=9\sqrt{3}). Step 3: Choosing a larger perfect square simplifies the answer in one step.

Step 1: (\sqrt{6}\times\sqrt{54}=\sqrt{324}). Step 2: (\sqrt{324}=18), so the result is rational. Step 3: After multiplication, check whether the inside number has become a perfect square.

Step 1: Since (9<11<16), (\sqrt{11}) lies between (3) and (4). Step 2: Its approximate value is (3.316), so (3.32) is close. Step 3: In estimation, first set the range using perfect squares.

Step 1: (216=36 \times 6). Step 2: (\sqrt{216}=\sqrt{36 \times 6}=6\sqrt{6}). Step 3: The remaining (6) has no perfect square factor, so the form is simplified.

Step 1: (36<39<49). Step 2: Therefore, (6<\sqrt{39}<7). Step 3: Nearby perfect squares give the best clues for comparing square roots.

Step 1: (\sqrt{28}=2\sqrt{7}), so (\sqrt{7}+\sqrt{28}=3\sqrt{7}). Step 2: (3\sqrt{7}) is irrational because (\sqrt{7}) is irrational. Step 3: In options, simplify first before deciding the nature of the result.

Step 1: The square root of a positive integer is rational only when the integer is a perfect square. Step 2: For example, (25) is a perfect square and (\sqrt{25}=5) is rational. Step 3: In square-root questions, identify perfect squares first.

Step 1: (\sqrt{2}\times\sqrt{8}=\sqrt{16}). Step 2: (\sqrt{16}=4), which is rational. Step 3: The product of two irrational numbers is not always irrational.

Step 1: (\sqrt{72}=6\sqrt{2}) and (\sqrt{18}=3\sqrt{2}). Step 2: (6\sqrt{2}-3\sqrt{2}=3\sqrt{2}). Step 3: Simplify both square roots before subtracting.

Step 1: (x-3=(3+\sqrt{5})-3). Step 2: This leaves (\sqrt{5}), which is irrational. Step 3: Simplify the expression before deciding the nature of the number.

Step 1: Terminating decimals and fractions are rational. Step 2: (\sqrt{18}=3\sqrt{2}), and (\sqrt{2}) is irrational. Step 3: Simplifying a square root often helps identify the number correctly.

Step 1: (\sqrt{12}=2\sqrt{3}), (\sqrt{27}=3\sqrt{3}), and (\sqrt{75}=5\sqrt{3}). Step 2: Adding gives (2\sqrt{3}+3\sqrt{3}+5\sqrt{3}=10\sqrt{3}). Step 3: Once radicals are like terms, add only the coefficients.

Step 1: Adding a rational number does not remove the irrational part. Step 2: For example, (2+\sqrt{3}) is irrational. Step 3: Be careful with always-type statements about sums or products of two irrational numbers.

Step 1: To simplify the denominator, multiply top and bottom by (\sqrt{5}). Step 2: (\frac{5}{\sqrt{5}}=\frac{5\sqrt{5}}{5}=\sqrt{5}). Step 3: Rationalising is useful when a square root appears in the denominator.

Step 1: Multiplying a square root by itself gives the number inside. Step 2: (\sqrt{a}\times\sqrt{a}=a), and if (a) is an integer, it is rational. Step 3: The square of an irrational number can be rational.

Step 1: Since (4<5<9), (2<\sqrt{5}<3). Step 2: (5) is not a perfect square, so (\sqrt{5}) is irrational. Step 3: Use nearby perfect squares to locate a square root.