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Simplifying Surds Slideshow 6, Mr Richard Sasaki, Room 307 Objectives β’ Understand the meaning of rational numbers β’ Understand the meaning of surd β’ Be able to check whether a number is a surd or not β’ Be able to simplify surds Rational Numbers What is a rational number? A rational number is a number that can be written in the form of a fraction where the numerator and denominator are integers. π π π₯ is rational if π₯ = where π, π β β€. π π If π₯ = where π, π β β€, we say π₯ β β. (π₯ is in the rational number set, β.) Rationality If a number is not rational, we say that it is irrational . π₯ is irrational if it canβt be written in the form π π where π, π β β€. Therefore, an irrational number π₯ β β. Example Show that 0.8 β β. 4 0.8 = 5. As 4, 5 β β€, β΄ 0.8 β β. π Note: If π₯ = π, π β 0 (You canβt divide a number by 0.) 0.45 = 9 20 and 9, 20 β β€. β΄ 0.45 β β. 9 = ±3 where -3 and 3 are integers. β΄ 9 β β€. β€ β β 0.25 = β0.5 = β 1 β 2 β β β€ where 1, 2 β β€. β΄ β 0.25 β β. π₯ = 0. 1, 10π₯ = 1. 1. β΄ 9π₯ = 1. 1 β 0. 1 = 1. βπ₯=1 9 π₯ = 0. 09, 100π₯ = 9. 09. β΄ 99π₯ = 9. 09 β 0. 09 = 9. β π₯ = 1 11 π₯ = 0. 5, 10π₯ = 5. 5. β΄ 9π₯ = 5. 5 β 0. 5 = 5. βπ₯=5 9 Surds What is a surd? A surd is an irrational root of an integer. We canβt remove its root symbol by simplifying it. Are the following surds? 2 9 Yes! No! 2 5 32 Yes! Yes! Even if the expression is not fully simplified, if it is a root and irrational, it is a surd. Multiplying Roots How do we multiply surds and roots? Letβs consider two roots, π and π where π₯ = π × π. If we square both sides, we getβ¦ 2 2 π₯ = π× π π₯2 = π × π × π × π 2 2 2 π₯ = π × π π₯2 = π × π If we square root both sides, we getβ¦ π₯ = π×π β΄ π × π β‘ π × π, where π, π β β. Simplifying Surds To simplify a surd, we need to write it in the form π π where π is as small as possible and π, π β β€. Note: Obviously, if π = 1, π π = π. Example Simplify 8. 8 = 4β2 = 4β 2 =2 2 We try to take remove square factors out and simplify them by removing their square root symbol. Answers - Easy Because 2 has positive and negative roots anyway. β΄ 2 2 β‘ ±2 2. 4 2 Yes 6 3 10 3 2 5 No 6 16 2 6 2 5 6 4 6 Yes 42 2 12 6 No, for example, 6 is a surd but 6 is not prime. Square numbers. 21 6 Answers β Hard (Questions 1 β 4) Let 16 be in the form π π where π, π β β€, π > 1, π β β€. 16 = ±4 or rather 4 1 in the form π π. As π = 1, 16 is not a surd. 8 3 9 10 2 93 24 3 ±22 17 5 β€ 10 15 14 2 39 3 Answers β Hard (Question 5) 288 1875 625 β‘ 144 β’ β‘ 72 β€ 125 36 β‘ β€ 25 92 = 2 2 23 = 2 23 β‘ 18 β€β€ 4 β‘ 9 1875 = 5 3 5 2 288 = 2 3 β’ β’ = 25 3 = 12 2 92 β‘ 46 β‘ 23