krit.club logo

Number - Surds and Radicals: Operations, Simplification, and Rationalization

Grade 8IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

A surd is an irrational number that can be expressed as a root of a rational number. For example, 2\sqrt{2} and 5\sqrt{5} are surds, but 9\sqrt{9} is not because it simplifies to 33.

•

Simplifying Surds: A surd is in its simplest form when the number under the radical has no perfect square factors (other than 11). This is done using the property a×b=a×b\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}.

•

Addition and Subtraction: You can only add or subtract 'like surds' (surds with the same number under the radical sign). For example, 23+53=732\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}, but 23+522\sqrt{3} + 5\sqrt{2} cannot be simplified further.

•

Multiplication and Division: Surds can be multiplied or divided regardless of whether they are like terms. Use the rules a×b=ab\sqrt{a} \times \sqrt{b} = \sqrt{ab} and ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}.

•

Rationalizing the Denominator: This is the process of eliminating a radical from the denominator of a fraction. For a fraction ka\frac{k}{\sqrt{a}}, multiply both the numerator and the denominator by a\sqrt{a} to get kaa\frac{k\sqrt{a}}{a}.

📐Formulae

ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b}

ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}

(a)2=a(\sqrt{a})^2 = a

ac±bc=(a±b)ca\sqrt{c} \pm b\sqrt{c} = (a \pm b)\sqrt{c}

1a=1×aa×a=aa\frac{1}{\sqrt{a}} = \frac{1 \times \sqrt{a}}{\sqrt{a} \times \sqrt{a}} = \frac{\sqrt{a}}{a}

💡Examples

Problem 1:

Simplify the surd 75\sqrt{75}.

Solution:

75=25×3=25×3=53\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}

Explanation:

Identify the largest perfect square factor of 7575, which is 2525. Use the product rule of radicals to separate them.

Problem 2:

Simplify 32+18−523\sqrt{2} + \sqrt{18} - 5\sqrt{2}.

Solution:

32+9×2−52=32+32−52=(3+3−5)2=12=23\sqrt{2} + \sqrt{9 \times 2} - 5\sqrt{2} = 3\sqrt{2} + 3\sqrt{2} - 5\sqrt{2} = (3 + 3 - 5)\sqrt{2} = 1\sqrt{2} = \sqrt{2}

Explanation:

First, simplify 18\sqrt{18} to 323\sqrt{2} so that all terms are 'like surds'. Then, combine the coefficients.

Problem 3:

Rationalize the denominator of 105\frac{10}{\sqrt{5}}.

Solution:

105×55=1055=25\frac{10}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{10\sqrt{5}}{5} = 2\sqrt{5}

Explanation:

Multiply the numerator and denominator by 5\sqrt{5} to remove the radical from the bottom, then simplify the resulting fraction.

Problem 4:

Expand and simplify (2+3)(2−3)(2 + \sqrt{3})(2 - \sqrt{3}).

Solution:

2(2)−23+23−(3)2=4−3=12(2) - 2\sqrt{3} + 2\sqrt{3} - (\sqrt{3})^2 = 4 - 3 = 1

Explanation:

This follows the algebraic identity (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2. Here a=2a=2 and b=3b=\sqrt{3}.