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Note for Surds and Indices - SI by Placement Factory

• Surds and Indices - SI
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Diagnostic test ___ ___ correct? A the square root of 4 is 2 or −2 B √4 = 2 __ __ A 30√6 __ . . 2 Convert 0.312 to a fraction. 104 78 39 B ____ C ___ A ____ 125 333 25 √ 9 ___ 16 C 26 D ___ 75 3 2_4 D −4.7 4 (3 √5 ) = B 225 __ B 8√2 __ A √30 C √150 D √900 C 7 D 7 + 4 √5 __ __ ___ A 7 B 7 + 2√10 C √14 D 2√7 __ ___ A 5 __ 1 B _8 A 8 A B 9 __ __ √15 B ____ 6 √5 C ___ 3 1 B _5 C √10 ___ √10 D ____ 6 1 __ √5 √ __ 16 102 = √40 __ = 7 ____ 8 __ A 4 √2 − 2 √5 √5 A ___ 6 ____ ___ ___ 1 4_9 __ __ __ __ ____ B √180 D 10 − 2√6 √5__ = 15 Expressed with a rational denominator, ____ 2√3 __ ____ __ __ C √10 − 4√3 14 (√5 + √2 ) 2 = D 4√2 6 Written in the form √n , 5√6 = ___ ___ ___ B √10 − 2 √6 __ __ C 16√2 __ __ ___ 5 In simplest form, √32 = __ ___ A √10 − 2 √3 __ __ D 9√5 C 15 ____ D √180 __ __ __ ___ A 2√8 ___ C 10√30 13 (2√3 + √5 )(2√3 − √5 ) = 2 A 45 __ B 30√2 12 √2 (√5 − 2√3 ) = 3 Which of the following is not a rational number? ___ __ __ D 2 √5 __ __ ___ D √−4 = −2 B B 4√2 11 2√3 × 5√6 = C −√4 = −2 A √11 __ A √20 __ __ C 2√3 + √2 __ ___ __ 10 √18 + √2 = 1 Which of the following statements is not ___ D √35 C 2√2 ___ __ 17 In index form, √k3 = 4 = 3 __ 4 __ A k 12 B k3 ___ √___3 37 ___ 1 1 B 2_3 __ A 18 ___ ___ __ A 8√15 B 10√10 − 2√5 C 8√5 D 2√15 + 6√10 ___ __ D k7 C 6 D 40.5 18 When evaluated, 273 = ___ 9 4√10 − 2√5 + 6√10 = C k4 2 __ 1 D 2_9 C 4_3 1 ___ D ____ √10 19 ___ −1 (__35) B 9 = 3 2 A 1_3 B −_5 5 C −_3 1 D __ 15 NUMBER & ALGEBRA The Diagnostic Test questions refer to the sections of text listed in the table below. 2 Question 1 2 3 4–8 9, 10 11–14 15 16–18 19 Section A B C D E F G H I Insight Maths 9 Australian Curriculum

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A Square root of a number The square root of a number, x, is the number that when multiplied by itself is equal to x. For example, the square root of 9 is 3 or −3, since 32 = 9 and (−3) 2 = 9. 11004 An aerial photo of a square __ house block (say 900 square metres). √x is the positive square root of x. __ __ __ For example, √9 = 3 and − √9 = −3. −√x is then the negative square root of x. Needs to be square. EXAMPLE 1 Find the following. a the square root of 81 b √8 1 a the square root of 81 = 9 or −9 b √81 = 9 __ ___ c −√81 ___ ___ c −√81 = −9 Exercise 11A 1 Find the following. a i the square root of 4 b i the square root of 25 c i the square root of 49 d i the square root of 64 ii ii ii ii __ 4 √___ iii iii iii iii 25 √___ 49 √___ √64 __ −√___ 4 −√___ 25 −√___ 49 −√64 ___ Since there is___ no number that when multiplied by itself is equal to −9, it is not possible to ﬁnd √−9 . We say that √−9 is undeﬁned. __ __ The square root of 0 is 0, since 0√0 = 0. Zero is neither positive nor negative but we deﬁne √0 = 0. NUMBER & ALGEBRA In general: __ • √x is undeﬁned for x < 0 __ • √x = 0 for x = 0 __ • √x is the positive square root of x when x > 0. __ • −√x is the negative square root of x when x > 0. EXAMPLE 2 Find the following, where possible. ___ a √36 ___ a √36 = 6 ___ b − √36 ___ b − √36 = −6 ____ __ c √−36 d √0 c undeﬁned d 0 Chapter 11 Surds and indices 3

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2 Find the following, where possible. ____ _____ a √100 b √−100 ___ ___ ____ f − √16 ____ g √−16 ____ l √−81 ____ k √−25 __ c √−4 h √−49 ____ m √−64 ___ d √0 __ i − √1 e √16 __ j √1 n −√100 o √−1 ____ ___ B Recurring decimals 3 1 As a decimal, _8 = 0.375 and _3 = 0.333 33 … 3 • When converted to a decimal, the fraction _8 terminates. That is, the digits after the decimal point stop after 3 places have been ﬁlled. We call this a terminating decimal. When converted to a decimal, all fractions either terminate or recur. 1 • When the fraction _3 is converted to a decimal, the digits after the decimal point keep repeating or recurring. We call this a recurring decimal. . 0.3333… is written 0. 3. We call this dot notation. The dot above the 3 indicates that this digit recurs. EXAMPLE 1 Write the following recurring decimals using dot notation. a 0.4444 … b 0.411 11 … d 0.415 415 415 … e 0.415 341 534 153 … . a 0. 4 . . d 0. 41 5 . b 0.4 1 . . e 0. 415 3 c 0.414 141 … .. c 0. 4 1 The dots are put above the ﬁrst and last digits of the group of digits that repeat. Exercise 11B 1 Write the following recurring decimals using dot notation. a 0.7777 … b 0.355 55 … c 0.282 828 … e 0.678 467 846 784 … f 1.4444 … g 6.922 22 … i 0.234 234 234 … j 0.033 33 … k 0.909 090 … m 0.217 77 … 11005 Photo of an aerial view of an NUMBER & ALGEBRA iron ore train or similar (say Pilbra) 4 Insight Maths 9 Australian Curriculum d 0.325 325 325 … h 0.494 949 … l 0.536 666 …

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EXAMPLE 2 Use your calculator to convert the following fractions to decimals. a 5 _ 8 b 2 _ 3 a By calculating 5 ÷ 8 or using the fraction key, _58 = 0.625. b By calculating 2 ÷ 3 or using the fraction key, the display could show 0.666666666 or 0.666666667, depending on the calculator used. . Both are approximations for the recurring decimal 0. 6. In the ﬁrst case the calculator has truncated the answer (because of the limitations of the display), and in the second case the calculator has automatically rounded up to the last decimal place. . 2 Hence _3 = 0. 6. 2 Convert the following fractions to decimals. a 7 _ 8 b f 1_23 g 5 _ 9 11 __ 18 c h 1 _ 6 22 __ 33 2 d ___ 11 5 e 1__ 12 13 i 1__ 22 j 11 __ 24 EXAMPLE 3 Convert the following decimals to fractions. a 0.8 b 0.63 c 8 4 _ a 0.8 = __ 10 = 5 148 c 0.148 = ____ 1000 63 b 0.63 = ___ 100 0.148 37 = ___ 250 3 Convert the following decimals to fractions. a 0.6 b 0.78 c d 0.08 0.125 e 0.256 EXAMPLE 4 Convert the following recurring decimals to fractions. . .. a 0. 4 b 0. 5 7 . Let n = 0. 4. n = 0.4444 … Then 10n = 4.4444 … By subtraction, 9n = 4 4 Hence n = __ 9 . 4 ∴ 0. 4 = _9 b .. Let n = 0. 5 7 n = 0.575757 … Then 100n = 57.575757 … By subtraction, 99n = 57 57 19 Hence n = ___ = ___ 99 33 .. 19 __ ∴ 0. 5 7 = 33 4 Convert the following recurring decimals to fractions. . . . a 0. 2 b 0. 3 c 0. 5 . d 0. 8 NUMBER & ALGEBRA a Before subtracting, multiply by the power of 10 that makes the decimal parts the same. . e 0. 7 . 5 Convert 0. 9 to a fraction. Discuss the result with your class. Chapter 11 Surds and indices 5