1.8

Surds

9 flashcards to master Surds

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Definition Flip

Define a surd. Provide an example.

Answer Flip

A surd is an irrational number that can be expressed as the root of a rational number. It cannot be simplified to a rational number.

Example: √2, √3, √5 are surds, but √4 = 2 is not.
Key Concept Flip

Simplify the surd: √75

Answer Flip

To simplify, find the largest perfect square factor of 75, which is 25. So, √75 = √(25 x 3) = √25 x √3 = 5√3.

Key Concept Flip

Explain how to rationalise the denominator of the fraction: 2/√3

Answer Flip

To rationalise, multiply both the numerator and denominator by the surd in the denominator. Thus, (2/√3) x (√3/√3) = 2√3/3.

Key Concept Flip

Rationalise the denominator: 5/(2 + √3)

Answer Flip

Multiply the numerator and denominator by the conjugate of the denominator (2 - √3). This gives: [5(2 - √3)] / [(2 + √3)(2 - √3)] = (10 - 5√3) / (4 - 3) = 10 - 5√3

Definition Flip

What is the conjugate of (√5 - 2)?

Answer Flip

The conjugate of a binomial expression containing a surd is found by changing the sign between the terms. Therefore, the conjugate of (√5 - 2) is (√5 + 2).

Key Concept Flip

Simplify: (3 + √2)(3 - √2)

Answer Flip

This is in the form (a+b)(a-b) = a² - b². Therefore, (3 + √2)(3 - √2) = 3² - (√2)² = 9 - 2 = 7.

Key Concept Flip

Explain why rationalising the denominator is important.

Answer Flip

Rationalising the denominator removes surds from the denominator, making it easier to compare and manipulate fractions. It also simplifies further calculations.

Key Concept Flip

Express √18 + √32 as a single surd.

Answer Flip

Simplify each surd first: √18 = √(9 x 2) = 3√2, and √32 = √(16 x 2) = 4√2. Then, 3√2 + 4√2 = 7√2.

Definition Flip

What is a 'radical' in the context of surds?

Answer Flip

A radical is the mathematical symbol (√) used to indicate a root, such as a square root or cube root. It signifies that a surd is the root of a number.

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1.7 Limits of accuracy 2.1 Algebraic notation and manipulation

Key Questions: Surds

Define a surd. Provide an example.

A surd is an irrational number that can be expressed as the root of a rational number. It cannot be simplified to a rational number.

Example: √2, √3, √5 are surds, but √4 = 2 is not.
What is the conjugate of (√5 - 2)?

The conjugate of a binomial expression containing a surd is found by changing the sign between the terms. Therefore, the conjugate of (√5 - 2) is (√5 + 2).

What is a 'radical' in the context of surds?

A radical is the mathematical symbol (√) used to indicate a root, such as a square root or cube root. It signifies that a surd is the root of a number.

About Surds (1.8)

These 9 flashcards cover everything you need to know about Surds for your Cambridge IGCSE Mathematics (0580) exam. Each card is designed based on the official syllabus requirements.

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