T1 Surds Notes + Set Work

Surds Overview

  • Definition: A surd is an expression containing a root, such as square root, that cannot be simplified to remove the root.

Learning Intentions

  • Rational vs Irrational: Ability to identify and categorize numbers as rational or irrational.

  • Simplification: Simplify surds using the highest square number factor.

  • Arithmetic with Surds:

    • Add and subtract like surds.

    • Multiply and divide surds.

  • Distributive Law: Apply when working with expressions involving surds.

  • Rationalization: Rationalize a denominator in fractions involving surds.

Mathematical Concepts

Number Sets

  • Natural Numbers (𝑁): Whole positive numbers.

  • Integers (𝑍): Whole positive and negative numbers.

  • Rational Numbers (𝑄): Numbers formed by dividing one integer by another.

  • Irrational Numbers (𝑄'): Numbers that cannot be expressed as a fraction of integers.

  • Real Numbers (𝑅): All rational and irrational numbers.

Basic Rules for Surds

  • If π‘Ž β‰₯ 0, then √(π‘Ž) is defined.

  • For any non-negative π‘Ž and 𝑏:

    • (π‘Ž οΏ½d7 𝑏) = √(π‘Ž) οΏ½d7 √(𝑏)

    • (π‘Ž Γ· 𝑏) = √(π‘Ž) Γ· √(𝑏)

Simplifying Surds

  • Find perfect square factors of numbers under square roots:

    • For example, √36 = 6 (since 36 is 6Β²).

  • Steps for Simplification:

    • Identify factors of the number.

    • Choose the largest perfect square.

    • Express the surd in its simplest form.

Entire Surds

  • Definition: Numbers made up entirely of surds.

  • Example: Express 3√2 + 2sqrt(10) as an entire surd.

Lesson Exercises

Operations with Surds

Addition/Subtraction

  • Can only add/subtract like terms (terms with the same surd).

  • Example Problems:

    • Simplify: √(3) + 2√(3) = 3√(3).

    • Simplify: 2√(3) - √(12) = 2√(3) - 2√(3) = 0.

Multiplication/Division

  • Use:

    • (π‘Žβˆšπ‘) Γ· (π‘βˆšπ‘‘) = (π‘Ž/c)√(b/d)

    • (π‘Žβˆšπ‘) οΏ½d7 (π‘βˆšπ‘‘) = (π‘Žc)√(𝑏d)

    • Practice Problems available.

Expansion Involving Surds

  • Distributive Law: a(b+c) = ab + ac

  • Example: Expand and simplify 2(√(7) - 2) = 2√(7) - 4.

Rationalizing Denominators

  • Importance: Easier calculations when surds are in the numerator.

  • Method: Multiply by the conjugate to eliminate the surd in the denominator.

  • Example:

    • If the denominator is 7 + √5, multiply by (7 - √5)/(7 - √5).

Setwork

  • Assignments to reinforce understanding:

    • Exercises on irrational numbers and simplifying surds (Page 312 improvements).

    • Addition/subtraction of surds and multiplication/division exercises (Pages 317, 322).

    • Expanding brackets with surds (Page 322).

    • Rationalizing the denominator (Page 327).

  • Review Material: Include multiple choice and extended response questions from Page 408.