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Number - Surds

Grade 9IGCSE

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

🔑Concepts

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Definition of a Surd: An irrational number that is expressed as a root of a rational number (e.g., √2, √3).

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Simplifying Surds: Factoring a number into a product of a perfect square and another number to simplify the radical.

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Like Surds: Surds with the same number under the root sign (the radicand) which can be added or subtracted.

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Multiplying and Dividing: Surds can be combined under a single root when multiplied or divided.

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Rationalizing the Denominator: The process of removing a surd from the bottom of a fraction by multiplying the numerator and denominator by an appropriate factor.

📐Formulae

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

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

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

ka=kaa\frac{k}{\sqrt{a}} = \frac{k\sqrt{a}}{a}

(a+b)(a−b)=a−b(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b

💡Examples

Problem 1:

Simplify 72\sqrt{72}.

Solution:

626\sqrt{2}

Explanation:

Find the largest perfect square factor of 72, which is 36. Rewrite as 36×2\sqrt{36 \times 2}. Since 36=6\sqrt{36} = 6, the expression simplifies to 626\sqrt{2}.

Problem 2:

Simplify 218+502\sqrt{18} + \sqrt{50}.

Solution:

11211\sqrt{2}

Explanation:

First, simplify each term: 218=29×2=2(32)=622\sqrt{18} = 2\sqrt{9 \times 2} = 2(3\sqrt{2}) = 6\sqrt{2}. Then, 50=25×2=52\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}. Adding them together: 62+52=1126\sqrt{2} + 5\sqrt{2} = 11\sqrt{2}.

Problem 3:

Rationalize the denominator of 63\frac{6}{\sqrt{3}}.

Solution:

232\sqrt{3}

Explanation:

Multiply both the numerator and the denominator by 3\sqrt{3}. This gives 6333=633\frac{6\sqrt{3}}{\sqrt{3}\sqrt{3}} = \frac{6\sqrt{3}}{3}. Simplify the fraction to get 232\sqrt{3}.

Problem 4:

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

Solution:

77

Explanation:

This is in the form (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2. Applying this: 32−(2)2=9−2=73^2 - (\sqrt{2})^2 = 9 - 2 = 7.

Problem 5:

Rationalize the denominator of 43−5\frac{4}{3 - \sqrt{5}}.

Solution:

3+53 + \sqrt{5}

Explanation:

Multiply the numerator and denominator by the conjugate of the denominator, which is (3+5)(3 + \sqrt{5}). Numerator: 4(3+5)=12+454(3 + \sqrt{5}) = 12 + 4\sqrt{5}. Denominator: (3−5)(3+5)=9−5=4(3 - \sqrt{5})(3 + \sqrt{5}) = 9 - 5 = 4. Result: 12+454=3+5\frac{12 + 4\sqrt{5}}{4} = 3 + \sqrt{5}.