Chapter 07 New Century Maths

Number/Patterns and algebra

Indices

7

The speed of light is about 300 000 km/s. In one year, light travels approximately 9 460 000 000 000 km. The light from the stars travels for many years before it is seen on Earth. Powers or indices provide a way to work easily with very large numbers or with very small numbers.

■ _{describe and evaluate numbers written in index form, using terms such as} ‘base’, ‘power’, ‘index’ and ‘exponent’

■ _{develop and use the index laws for multiplying and dividing terms with the} same base, and for the power of a base raised to a power

■ _{develop and use zero and negative indices}

■ _{use fractional indices for square roots and cube roots} ■ _{express and order numbers in scientific notation}

■ _{convert numbers expressed in scientific notation to decimal form} ■ _{enter and read scientific notation on a calculator}

■ _{calculate with numbers expressed in scientific notation.}

■ _{base A number that is raised to a power, meaning that it is multiplied by} itself repeatedly. For example, in 25_{, the base is 2.}

■ _{power The number of times a base is multiplied by itself. For example, 2}5 means 2 × 2 × 2 × 2 × 2, and is 2 to the power of 5. A power is also called an index or an exponent.

■ _{index notation or index form Repeated multiplication written in the form}

an_{. For example, 2} × ₂ × ₂ × ₂ × _{2 written using index notation is 2}5_. ■ _{negative power A power that is a negative number, as in the}

expression 3−2_.

■ _{scientific notation A shorter way of writing very large or very small} numbers using powers of 10. For example, 9 460 000 000 000 in scientific notation is 9.46 × 1012_.

The story is that Sissa ben Dahir, who invented chess, was offered any reward he wanted by the Indian King Shirham. Sissa asked for the following:

‘I will have one grain of wheat for the first square of my chessboard, two grains of wheat for the second, four for the third and so on to the sixty-fourth square.’

King Shirham granted his request without thinking!

■ _{How many grains of wheat would be needed for the 64th square?} ■ _{How many grains of wheat would be needed altogether to meet Sissa’s}

request?

■ _{If a grain of wheat weighs 100 mg, how many tonnes of wheat would there} be on the chessboard?

In this chapter you will:

Wordbank

Powers

The numbers 2, 4, 8, 16, … are powers of 2. (They can also be written as 21_{, 2}2_{, 2}3_{, 2}4_{,… .)}

Similarly, the numbers 3, 9, 27, 81, 243, … are powers of 3.

1 Evaluate: a 4 × 4 b 2 × 2 × 2 × 2 × 2 c 3 × 3 d 5 × 5 × 5 e 10 × 10 × 10 × 10 × 10 f 7 × 7 g 4 × 4 × 4 h 8 × 8 × 8 × 8 × 8 × 8 i 12 × 12 × 12 × 12 2 Evaluate: a 43 _{b 10}2 _{c 2}6 _{d 3}4 _{e 4}2 _{f 10}6

3 Express in index form:

a 5 × 5 b 4 × 4 × 4 × 4 × 4 c 6 × 6 × 6

d 3 × 3 × 3 × 3 × 3 × 3 e y × y f m × m × m × m × m

g a × a × a h x × x × x × x × x × x i d × d × d × d

4 Write in expanded form:

a 103 _{b 8}2 _{c 1}5 _{d 2}4 _{e 3}1 f k2 _{g w}4 _{h d}5 _{i p}1 _{j c}3 5 Evaluate: a b c d e 6 Evaluate: a b c d e f 400 289 1024 225 625 8 3 3 ₂₇ 3 _-8 3 _-216 3₁₀₀₀ 3_-27

Start up

Worksheet 7-01 Brainstarters 7

Working mathematically

Reasoning and reflecting: Powers and the power key

Numbers expressed as powers of numbers, such as 27_{, can be easily evaluated using the}

power key ( or or ) on your calculator.

1 a Evaluate 24= ₂ × ₂ × ₂ × ₂ = _?

b Evaluate 24 _{using the power key on your calculator as follows:}

2 4

(Note: Your answers for parts a and b should be the same.)

2 Use the power key to evaluate each of the following. Compare your answers to those of

other students.

a 45 _{b 7}7 _{c 3}4 _{d 11}8

3 a Copy the table below into your book and use your calculator to evaluate the first six

powers of 4, 5, 6 and 7, and enter them in your table. Compare your results with those of other students in your class.

^

x

y

_y

x

^

=

Index notation

Consider 2 × 2 × 2 × 2 × 2 = 25_.

2 × 2 × 2 × 2 × 2 is the expanded or factor form. 25 _{is the index notation or exponent form.}

In 25_{, 5 is called the index or the power or the exponent.}

The base is 2.

25 _{is read as ‘2 to the power of 5’ or ‘2 to the 5th’.} b Using the results in your table, evaluate:

i 81 _{ii 15}1 _{iii 2}1 _{iv 23}1

c What is the value of a1 _{(that is, any number to the power of 1)?}

4 a Evaluate the powers of 2 (21_{, 2}2_{, 2}3_{, …). What is the largest power of 2 that your}

calculator can display as a whole number?

b Find the largest power of each of the following numbers that your calculator can

display as a whole number.

i 3 ii 4 iii 5 iv 6 v 7

Compare your results with those of other students in your class.

Powers of 4 Powers of 5 Powers of 6 Powers of 7

41= ₅1= ₆1= ₇1= 42= ₅2= ₆2= ₇2=



46= ₅6= ₆6= ₇6= Skillsheet 7-01 Indices

2

5

index, power or exponent base SkillBuilder 11-01 Introduction to indices

Example 1

Express in index form:

a 3 × 3 × 3 × 3 b m × m × m × m × m c a × a × … × a

Solution

a 3 × 3 × 3 × 3 = 34 _{b m} × _m × _m × _m × _m = _m5 _{c a} × _a × _… × _a = _an

Express in index form:

a 5 × 5 × 5 × 6 × 6 × 6 × 6 b p × p × p × p × t × t × t × t × t × t c a × a × … × a × b × b × … × b

Solution

a 5 × 5 × 5 × 6 × 6 × 6 × 6 = 53× ₆4 b p × p × p × p × t × t × t × t × t × t = p4× _t6= _p4_t6        n factors        4 factors            5 factors        n factors

Example 2

       n factors        m factors 3 factors        4 factors             4 factors 6 factors         

c a × a × … × a × b × b × … × b = an× _bm= _an_bm

Express in expanded form:

a 35 _{b 5x}3_m4

Solution

a 35= ₃ × ₃ × ₃ × ₃ × _{3 b 5x}3_m4= ₅ × _x × _x × _x × _m × _m × _m × _m        n factors        m factors

Example 3

1 For each of the following:

i state the base ii state the index iii write the expression in words.

a 37 _{b 7}3 _{c k}4 _{d 4}k _{e a}n

2 Express in index form:

a 5 × 5 × 5 × 5 b 10 × 10 c 8 × 8 × 8 d 32 × 32

e 9 × 9 × 9 × 9 × 9 f 12 × 12 × 12 g 1 × 1 × 1 × 1 h 6 3 Write using index notation:

a a × a × a × a b m × m c y × y × y × y

d q × q × q × q × q × q e p × p × p f w

4 Write these in index notation:

a 3 × 3 × 2 × 2 × 2 × 2 × 2 b 3 × 3 × 3 × 3 × 7 × 7 × 7 c 5 × 5 × 5 × 5 × 5 × 5 × 8 × 8

d 6 × 6 × 6 × k × k e x × y × x × y × x f 5 × n × 5 × n × n

5 Write in expanded or factor form:

a 64 _{b 10}3 _{c 6}4× ₁₀3 _{d p}4

e 5p4 _{f 5}2_p4 _{g p}4_q5 _{h 5p}4_q5

i 52_p4_q5 _{j ab}3 _{k ab}3_c2 _{l a}4_bc2

m m3_n4 _{n 2y}3_d2 _{o 4}2_a3_{m p w}4_y2_v3

6 Evaluate the following.

a 24 _{b 3}3 _{c 5}2 _{d 4}3

e 27 _{f 5}3 _{g 13}2 _{h 8}3

i 64 _{j 7}3 _{k 2}10 _{l 3}5

m 52× ₅5 _{n 3}3× ₁₀4 _{o 4}4× ₆2 _{p 3}5× ₅3

7 Evaluate, correct to three decimal places:

a 3.17 _{b (0.145)}2 _{c d (}−_2.5)7

e (1.1)5 _{f g h (0.18)}2 8 Find the missing powers in:

a 8 = 2? _{b 81} = ₃? _{c 216} = ₆? _{d 144} = ₁₂?

e 4096 = 2? _{f 2401} = ₇? _{g 64} = ₂? _{h 625} = ₅? 9 Evaluate, correct to 2 significant figures:

a 126 _{b (}−₁₁₎5 _{c 2}12 d (3.1)3 _{e (}−_1.11)2 _{f (7.2)}4 −2 5 ---   4 12 7 ---   4 −2 3 ---   5

Exercise 7-01

Example 1 Example 2 Example 3 CAS 7-01 Index form

10 If p = 4, q = 3, r = 5, evaluate the following.

a p4 _{b pq}3 _{c (pq)}3

d pq2_{r e (pqr)}2 _f

11 a Evaluate the terms of the pattern (−1)1_{, (}−₁₎2_{, (}−₁₎3_{, (}−₁₎4_{, (}−₁₎5_{, (}−₁₎6_{, …}

Write down what you observe about the odd and even powers and the sign of the answer.

b Without using a calculator, find the value of:

i (−1)98 _{ii (}−₁₎99

c Predict the values of the following.

i (−1)n _{when n is even ii (}−₁₎n _{when n is odd} d Hence evaluate:

i (−1)1+ ₍−₁₎2+ ₍−₁₎3+ ₍−₁₎4+ _… + ₍−₁₎36 ii (−1)1+ ₍−₁₎2+ ₍−₁₎3+ ₍−₁₎4+ _… + ₍−₁₎37 12 a Evaluate the following.

i 62 _{ii 66}2 _{iii 666}2 _{iv 6666}2

b Predict the values of the following.

i (666 666)2 _{ii (666 666 666)}2 q r ---   3

Working mathematically

Applying strategies and reasoning: Cell growth

Use a spreadsheet to help you with this investigation.

Over the centuries, millions of people have contracted diseases such as smallpox, typhoid and diphtheria. These diseases start off as a few cells that multiply at an alarming rate until there are too many in the body, causing the person to become ill. In some cases this can be fatal.

Suppose one of these diseases grows by the cells splitting into equal parts every 10 seconds; that is, every 10 seconds, the number of cells doubles.

Disease A

1 Starting with one cell, calculate the cell population after:

a 30 s b 40 s c 1 min d 1 min 30 s

e 2 min f 3 min g 4 min h 4 min 20 s

t = 0 s 1 cell 2 cells 4 cells t = 10 s t = 20 s Spreadsheet 7-01 Cell growth

The index laws

2 Starting with one cell, find how long it will take until there are:

a 64 cells b 256 cells

c over 500 cells d over 1000 cells e over 3000 cells f over 10 000 cells

g over 1 000 000 cells h over 40 000 000 cells

Further diseases are discovered that multiply at different rates:

• Disease B: A single cell divides into three identical cells every 15 seconds. • Disease C: A single cell divides into four identical cells every 20 seconds. • Disease D: A single cell divides into five identical cells every 30 seconds.

3 Expand your spreadsheet to show all four disease strains, and answer Questions 1 and 2

for strains B, C and D.

4 If the diseases all begin from one cell at time 0 seconds, when does the growth of each

strain pass that of the others?

5 Graph the growth of each disease on your graphics calculator or by using the Graph

option in the spreadsheet.

6 In your own words, describe in writing the shape of the exponential graph generated for

each disease.

Working mathematically

Reasoning and reflecting: Multiplying terms with the same base

1 Use a calculator to find the value of:

a i 23× ₂4 _{ii 2}7 _{b i 3}5× ₃3 _{ii 3}8

c i 44× ₄2 _{ii 4}6 _{d i 5}6× ₅3 _{ii 5}9

2 What do you notice about each pair of answers in Question 1? 3 Is it true that 25× ₂7= ₂12_{? Explain.}

4 State whether each of the following are true (T) or false (F).

Explain each choice.

a 26× ₂4= ₂10 _{b 7}4× ₇8= ₇32 _{c 4}5× ₄8= ₄40 _{d 3}7× ₃12= ₃19 5 Copy and complete the following.

a 47× ₄3= ₄… _{b 5}3× ₅4= ₅… _{c 6}8× ₆5= ₆… d 83× ₈2= _{… e k}3× _k8= _k… _{f m}3× _m7= _… 6 Use a calculator to find the value of:

a i 23× ₂5 _{ii 4}8 _{b i 5}4× ₅6 _{ii 25}10

c i 37× ₃4 _{ii 9}11 _{d i 6}2× ₆3 _{ii 36}5

7 Use your results from Question 6 to decide whether these are true (T) or false (F): a 23× ₂5= ₄8 _{b 5}4× ₅6= ₂₅10 _{c 3}7× ₃4= ₉11 _{d 6}2× ₆3= ₃₆5 8 Write true (T) or false (F) for each of the following.

a 53× ₅8= ₂₅11 _{b 2}7× ₂10= ₂17 _{c 7}3× ₇2= ₇5 d 43× ₄10= ₄30 _{e 5}3× ₅4= ₂₅7 _{f 3}3× ₃9= ₃12

Law 1: Multiplying terms with the same base

Consider 24× ₂3= ₍₂ × ₂ × ₂ × ₂₎ × ₍₂ × ₂ × ₂₎ = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 27 But 24 + 3= ₂7 ∴ 24× ₂3= ₂4 + 3= ₂7

Proof:

am× _an = = = am + n SkillBuilder 11-02

The first index law

When multiplying terms with the same base, add the powers:

am× _an= _am + n a_{      }×a×…×a×a_{      }×a×…×a n factors m factors a×a×…×a m+n factors       

Example 4

Simplify the following, expressing your answers in index form.

a 63× ₆7 _{b 5} × ₅3 _{c y}4× _y8

Solution

Simplify the following.

a 3p4× _2p6 _{b 5e}2f × _3e4_f5

Solution

a 63× ₆7= ₆3 + 7 = 610 b 5 × 53= ₅1× ₅3 = 51 + 3 = 54 c y4× _y8= _y4 + 8 = y12 a 3p4× _2p6= ₍₃ × ₂₎ × _(p4× _p6₎ = 6p4 + 6 = 6p10 b 5e2f × _3e4_f5= ₍₅ × ₃₎ × _(e2× _e4₎ × _(f × _f5₎ = 15e2 + 4_f1 + 5 = 15e6_f6

Example 5

1 Simplify (giving answers in index notation):

a 103× ₁₀2 _{b 10} × ₁₀4 _{c 3}2× ₃5 _{d 7}4× ₇ e 8 × 83× ₈4 _{f 5}4× ₅ × ₅4 _{g 6} × ₆2× ₆3× ₆4 _{h 4}4× ₄4× ₄4 i 117× ₁₁13 _{j 2} × ₂3 _{k 3}4× ₃ × ₃7 _{l 7}2× ₇5× ₇ 2 Simplify: a x × x4 _{b g}4× _g4 _{c w}7× _{w d b}3× _b10 e p10× _p10 _f r × _{r g y} × _y3× _y2 _{h m}3× _m × _m4

Exercise 7-02

CAS 7-02 Index multiplication Example 4

Law 2: Dividing terms with the same base

Consider 57÷ ₅4= = = 5 × 5 × 5 = 53 3 Simplify: a 3p2× _2p5 _{b 4y}10× _3y2 _{c 6m} × _3m8 d h3× _5h8 _{e 3q}3× _8q8 _{f 2a}2× _5a5 g 5n8× _6n8 _{h 2b}3× _15b6 _{i 3e}4× _e6 j 10p3× _5p2_{q k 8a}3_b2× _2b3_{a l 4w}5_y2× _5w4_y3 m 5a3_c2× _2b4_{c n 10p}3_q8× _qp2 _{o 4g}3_h2× _5gh4 4 Write true (T) or false (F) for each of the following.

a 53× ₃7= ₁₅10 _{b 7}2× ₈2= ₅₆4 _{c 3} × ₇2= ₂₁2 _{d 4}3× ₄7= ₄10 e 32× ₂4× ₃2× ₂5= ₃4× ₂9 _{f 5}2× ₅3= ₂₅15 _{g 2}7× ₂8= ₂15

h 73× ₇5= ₄₉8 _{i 4}2× ₃3= ₁₂6 _{j 5}4× ₃2× ₃7× ₅ = ₃9× ₅5 5 Simplify and evaluate:

a 23× ₂5 _{b 2}3× ₅2 _{c 10}2× ₂10 _{d 5}3× ₃5 e 33× ₃3 _{f 5}3× ₂3 _{g 10}2× ₁₀3 _{h 2}10× ₁₀3 6 Simplify: a x4× _x3× _x2 _{b y}6× _x3× _{y c 5} × _3n × _4n2 _{d 5} × _m × _4n2 e 5qp × 4q2× _5p3 _{f (a}4× _b3₎ × _(a4× _b2_{) g 4}a× ₄b _{h 2}x + 1× ₂x i 32y× ₃y _{j (p} + _q)2× _(p + _q)3 _{k (x – y)} × _{(x – y)}2 _{l (a} + ₃₎n× _(a + ₃₎ Example 5 SkillBuilder 11-06 Multiplying terms with indices

Working mathematically

Reasoning and reflecting: Dividing terms with the same base

1 Use a calculator to find the value of:

a i 210÷ ₂7 _{ii 2}3 _{b i 5}5÷ ₅3 _{ii 5}2

c i 37÷ ₃2 _{ii 3}5 _{d i 6}8÷ ₆4 _{ii 6}4

2 What do you notice about each pair of answers? 3 Is it true that 38÷ ₃6= ₃2_{? Explain.}

4 State whether each of the following are true (T) or false (F).

a 310÷ ₃6= ₃4 _{b 4}8÷ ₄2= ₄4 _{c 2}12÷ ₂3= ₂4 _{d 6}10÷ ₆5= ₆5 5 Copy and complete the following.

a 27÷ ₂3= ₂… _{b 5}8÷ ₅6= ₅… _{c 6}7÷ ₆2= ₆… d 311÷ ₃6= _{… e y}8÷ _y5= _{… f m}12÷ _m10= _… 6 Write true (T) or false (F) for the following.

a 106÷ ₁₀2= ₁₀4 _{b 10}6÷ ₁₀2= ₁3 _{c 10}6÷ ₁₀2= ₁₀3 d 812÷ ₈3= ₈9 _{e 4}10÷ ₄5= ₄2 _{f 6}6÷ ₆2= ₆4 SkillBuilder 11-03 Division of terms with indices 57 54 ---5×5×5×5×5×5×5 5×5×5×5 ---1 1 1 1 1 1 1 1

But 57 − 4= ₅3 ∴ 57÷ ₅4= = ₅7 − 4= ₅3

Proof:

am÷ _an= = = a × a × … × a(m − n factors) = am − n 57 54

---When dividing terms with the same base, subtract the powers:

am÷ _an=am = _am − n an ---am an ---a×a×a×a×a×…×a a×a×a×a×…×a --- (m factors) (n factors) 1 1 1 1

Example 6

Simplify the following, expressing your answers in index form.

a 45÷ ₄3 _{b c y}12÷ _y3

Solution

a 45÷ ₅3= ₄5 − 3 _b = ₁₀7 − 4 _{c y}12÷ _y3= _y12 − 3

= 42 = ₁₀3 = _y9

Simplify the following.

a k7÷ _{k b 15m}8÷ _3m2 _{c 30a}5_b7÷ _10a2_b5

Solution

a k7÷ _k = _k7÷ _k1 = k7 − 1 = k6 b 15m8÷ _3m2= = 5m8 − 2 = 5m6 c 30a5_b7÷ _10a2_b5= = 3a5 − 2_b7 − 5 = 3a3_b2 107 104 ---107 104

---Example 7

15m8 3m2 ---5 1 30a5b7 10a2b5 ---1 3

Just for the record

Remember that taxi

Indian mathematician Srinivasa Ramanujan (1888–1920) loved working with numbers. One day he was visited by a friend in a taxi numbered 1729. When Ramanujan heard the number, he immediately said ‘1729 is a very interesting number as it is the smallest number that can be expressed as the sum of two cubes in two different ways.’

This means that we can write 1729 = x3+ _y3

Here is one of the possible ways:

1729 = 103+ ₉3

1 Simplify, giving your answers in index form: a b c d 74÷ ₇3 e 105÷ ₁₀5 _{f 8}5÷ _{8 g 20}15÷ ₂₀5 _h i 680÷ ₆20 _{j 8}15÷ ₈11 _{k 3}12÷ ₃6 _{l 2}18÷ ₂11 2 Simplify: a b c d b16÷ _b15 e m16÷ _m16 _{f n}7÷ _{n g t}18÷ _t9 _h i e30÷ _e10 _{j k p}15÷ _p10 _{l w}24÷ _w6

3 Simplify the following.

a 10y15÷ _5y3 _{b 20w}9÷ _4w3 _{c 24r}8÷ _3r2 _d

e f g 16h10÷ _{8h h 15y}8÷ _15y4

i 18g60÷ _6g4 _{j k l}

m 20m15_n ÷ _2m14 _{n 36y}8_x7÷ _12x3_{y o 44e}4_f10÷ _4ef2 _{p 30k}7_m4÷ _6k6_m2 4 Write true (T) or false (F) for the following.

a 103÷ ₂2= ₅1 _{b 8}4÷ ₄4= ₂2 _{c 12}10÷ ₁₂10= _{1 d 15}8÷ ₁₅4= ₁₅2 e 109÷ ₁₀3= ₁₀6 _{f 7}4÷ ₇2= ₁2 _g ÷ ₄2 _{h 12}3÷ ₃3= ₄1 5 Evaluate: a 210÷ ₂5 _{b 4}5÷ ₂3 _{c 3}3÷ ₂3 _{d e f} g 45÷ ₂10 _{h 20}3÷ ₅3 _{i 10}6÷ ₅4 _{j 4}9÷ ₈3 _{k l 3}10÷ ₂₇2 6 Simplify: a b c d 4a÷ ₄b _{e 2}x + 1÷ ₂x _{f 3}2y÷ ₃y g h i × 58 52 --- 9 12 93 --- 2 27 23 ---220 2 ---h20 h4 --- y 8 y2 --- a 12 a4 ---w25 w ---d9 d5 ---30x4 x3 ---10m10 2m2 --- 12g 12 6g6 ---a6b3 a2b2 --- 36 p 8_q3 4 p4q --- 100 f 2_g4 5 f g2 ---204 52 ---103 23 --- 5 4 54 --- 2 10 52 ---125 68 ---x4×x3 x2 --- y10 y3×y --- a5×a3 a×a4 ---4m3×5m7 10m6 --- 6n 16_×8n4 3n3×4n5 --- p 6 6 p2 --- 30 p 4 5 p

---Exercise 7-03

Example 6 Example 7 CAS 7-03 Index division SkillBuilder 11-04

Using the second index law

Working mathematically

Reasoning and reflecting: Powers to powers

1 Use a calculator to find the value of:

a i (23₎2 _{ii 2}6 _{b i (3}4₎3 _{ii 3}12

c i (52₎3 _{ii 5}6 _{d i (2}5₎4 _{ii 2}20

Law 3: Raising a power to a power

Consider (42₎5= ₄2× ₄2× ₄2× ₄2× ₄2 = (4 × 4) × (4 × 4) × (4 × 4) × (4 × 4) × (4 × 4) = 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 = 410 But 42 × 5= ₄10 ∴ (42₎5= ₄2×5= ₄10

Proof:

Law 4: Powers of products and quotients

Consider (2 × 3)5= ₍₂ × ₃₎ × ₍₂ × ₃₎ × ₍₂ × ₃₎ × ₍₂ × ₃₎ × ₍₂ × ₃₎ = 2 × 2 × 2 × 2 × 2 × 3 ×3 × 3 × 3 × 3

= 25× ₃5

3 Is it true that (27₎3= ₂21_{? Explain.}

4 State whether each of the following is true (T) or false (F).

a (35₎3= ₃15 _{b (2}3₎2= ₂5 _{c (2}10₎4= ₂14 d (42₎5= ₄10 _{e (3}3₎6= ₃18 _{f (5}2₎4= ₅6 5 Copy and complete:

a (37₎2= ₃… _{b (5}2₎6= ₅… _{c (4}5₎2= ₄…

d (a3₎4= _a… _{e (8}3₎7= _{… f (k}4₎6= _…

6 State whether the following are true (T) or false (F).

a (25₎7= ₂12 _{b (2}8₎3= ₂24 _{c (5}3₎4= ₅12 d (73₎7= ₇7 _{e (8}4₎5= ₈9 _{f (6}6₎5= ₆30

When raising a term with a power to another power, multiply the powers: (am₎n= _am × n

(am₎n= _am× _am× _… × _am

= a × a × … × a × a × a × … × a × … × a × a × … × a

= am × n n factors

m factors m factors m factors

n lots of m factors

Example 8

Simplify the following, expressing your answers in index form.

a (23₎5 _{b (y}2₎14

Solution

a (23₎5= ₂3 × 5 = 215 b (y2₎14= _y2 × 14 = y28 SkillBuilder 11-07 Multiplying expressions with brackets

Also = = =

Proof:

(ab)n= = a × b × a × b × … × a × b = = an× _bn Also = × × … × = = 5 3 ---   4 ₅ 3 --- 5 3 --- 5 3 --- 5 3 ---× × × 5×5×5×5 3×3×3×3 ---54 34

---Powers of products and quotients:

(ab)n= _an_bn _and a b ---   n an bn ---= ab×ab×…×ab n factors          a×a×…×a n factors b×b×…×b n factors        ×       a b ---   n a b --- a b --- a b ---n factors          a×a×…×a b×b×b×…×b --- (n factors) n factors ( ) an bn

---Example 9

Simplify each of the following.

a (2k)5 _{b (5m}4₎3 _{c d}

Solution

a (2k)5= ₂5× _k5 = 32k5 b (5m4₎3= ₅3× _(m4₎3 = 125 × m4 × 3 = 125m12 c = = d = = = m 4 ----   3 _2w3 3 ---   4 m 4 ----   3 _m3 43 ---m 64 ---3 2w3 3 ---   4 _(2w3₎ 34 ---4 24×( )w3 4 34 ---16w12 81

---Just for the record

The house fly

The female common house fly, Musca domestica, can lay up to 1000 eggs at a time. In three weeks these reach maturity and are ready to breed. Huge populations would result if all the descendants of a single pair of house flies survived and reproduced. Fortunately, this is not the case as the mortality rate is very high. The few house flies we see are the true survivors.

Over the 13 weeks of summer, how many descendants could a single pair of house flies produce, assuming that each pair (original and descendants) mates only once?

(Give your answer in index form.)

1 Simplify, giving your answers in index form:

a (43₎2 _{b (5}2₎8 _{c (3}3₎4 _{d (2}7₎4 _{e (2}1₎2

f (94₎3 _{g (10}0₎2 _{h (6}4₎5 _{i (5}3₎5 _{j (2}5₎10

k (31₎5 _{l (7}3₎0 _{m (2}2₎10 _{n (13}2₎2 _{o (4}4₎4 2 Simplify each of the following. Give your answers in index form.

a (e2₎4 _{b (t}5₎5 _{c (y}3₎7 _{d (c)}5 _{e (m}7₎5

f (y4₎4 _{g (h}0₎6 _{h (p}6₎3 _{i (w}4₎1 _{j (x}1₎10

k (n3₎8 _{l (d}3₎3 _{m (k}5₎10 _{n (d}3₎4 _{o (a}8₎8

3 Simplify the following.

a (2d )4 _{b (5m)}2 _{c (4y}5₎2 _{d (3x}2₎4 _{e (5m}6₎5

f (2w5₎3 _{g (10d}5₎4 _{h (3e}7₎3 _{i (2b}4₎1 _{j (6d}6₎2 k (3f4₎5 _{l (2c}3₎10 _{m (3h}5₎4 _{n (6k)}2 _{o (8w}3₎2 4 Simplify each of the following.

a b c d

e f g h

i j k l

5 Simplify the following, giving your answers in index form.

a (m3₎10 _{b (5t)}3 _{c (}−₂₎8 _{d (}−x)3 e (y3₎12 _{f (4w}5₎4 _{g (}−_2d)5 _{h (2}10₎10 i (−3p2₎3 _{j (}−_5m3₎2 _{k (3f}5₎5 _{l (}−m2₎4 6 Evaluate: a (23₎2 _{b (}−₃2₎2 _{c (10}2₎3 _{d (}−₅₎2 e (−2)3 _{f (}−₄2₎3 _{g (}−₃4₎2 _{h (}−₅2₎3 i j k l e 2 ---   5 _x 7 ---   2 _3m 2 ---   3 _5h 6 ---   2 f2 3 ---   4 n5 p2 ---     8 w2 t3 ---     5 am c ---   4 2k3 5 ---   2 3r4 c2 ---     2 a2b d5 ---     4 5c2 3 x3 ---     3 3 2 ---   2 ₂ 5 ---   2 ₅ 2 ---   3 −3 4 ---   2

Exercise 7-04

Example 8 Example 9 Worksheet 7-02 Indices puzzle SkillBuilder 11-08

The fourth index law

7 Simplify each of the following. a (l3_m5₎6 _{b (x}2_y4₎5 _{c d} e (−we2_k3₎3 _{f g h (w}2_ak4₎8 i (−3m2_n)5 _{j (}−_2p2_w3₎4 _{k (a}2_d3_y5₎0 _l m n o (4k3_m)3 _{p (8k}4_y5₎2 q (3a3_df4₎5 _{r (6d}5_p2₎4 _{s t} 8 Simplify: a (52₎x _{b (5}x₎2 _{c ((x} + ₁₎2₎3 d (b3₎4÷ _(b4₎2 _{e (b}3₎4× _(b4₎2 _{f (6n}4₎2× _(3n2₎2 g (6n4₎2÷ _(3n2₎2 _{h i} a4 d3 ---     7 −m2 n ---   4 2 y3 x2 ---     3 p2q3 t4 ---     5 a2c4 d7 ---     0 d2e5f 4 ---   3 −m2n3 2 y5 ---     5 −3a y4 b2 ---     5 k p3 3q4 ---     2 3 x5 ( )3 x2 ( )4 --- 5 y 5 2 y ( )5 × 8 y2×( )y32

---Working mathematically

Questioning and reasoning: The power of zero

1 Copy and complete the following sentence.

A number remains unchanged when multiplied or divided by …

2 Copy and complete the following.

a 34× ₃0= ₃? _{b 5}2× ₅0= ₅? _{c 2}0× ₂7= ₂? _{d 7}0× ₇3= ₇? e 45× ₄0= ₄? _{f 5}0× ₅7= ₅? _{g 3}0× ₃5= _{? h 8}0× ₈6= _? 3 Copy and complete the following.

a 25÷ ₂0= ₂? _{b 3}5÷ ₃0= ₃? _{c 4}2÷ ₄0= ₄? _{d 9}3÷ ₉0= ₉? e 56÷ ₅0= ₅? _{f 8}4÷ ₈0= ₈? _{g 15}7÷ ₁₅0= _{? h 6}8÷ ₆0= _? 4 Copy and complete the following tables. Compare your answers with those of other

students.

5 Look at your results from Questions 1 to 4. Can you suggest a value for any number (or

base) raised to the power of zero (for example, 30= _{?, 5}0= _{?)? Explain.} a Index form Number b Index form Number c Index form Number 25 _{32 3}5 _{243 8}5 24 _{16 3}4 ₈4 23 ₃3 ₈3 22 ₃2 ₈2 21 ₃1 ₈1 20 ₃0 ₈0

The zero index

Consider 53÷ ₅3= = = 1 But 53÷ ₅3= ₅3 − 3= ₅0 ∴ 50= ₁

Proof:

am÷ _am= = 1 but am÷ _am= _am − m= _a0 ∴ a0= ₁ SkillBuilder 11-05 Raising to the power of 0 53 53 ---5×5×5 5×5×5 ---1 _{1 1} 1 1 1

Any number raised to the power of zero is equal to 1:

a0= ₁ a×a×a×…×a a×a×a×…×a ---1 1 1 1 1 1 1 1 (m factors) (m factors)

Example 10

Simplify the following.

a 70 _{b (}−₃₎0 _{c m}0

Solution

a 70= _{1 b (}−₃₎0= _{1 c m}0= ₁ Simplify: a (ab)0 _{b (5k)}0

Solution

a (ab)0= _{1 b (5k)}0= ₁ Simplify: a 5d0 _{b (3y)}0+ _3y0

Solution

a 5d0= ₅ × _d0 = 5 × 1 = 5 b (3y)0+ _3y0= ₁ + ₃ × _y0 = 1 + 3 × 1 = 4

Example 11

Example 12

1 Simplify the following. a 80 _{b (}−₂₎0 _{c d}0 _{d m}0 e f (−6)0 _{g (}−₇₀₀₎0 _{h (1 000 000)}0 i (−14)0 _{j k a}0 _l 2 Simplify: a (km)0 _{b (x}2_y)0 _{c (xyw)}0 _{d (}−ab)0 e f g (7y)0 _{h (9cd)}0

3 Simplify the following.

a 70+ ₂0 _{b 3y}0 _c −_(4m)0 _{d 3} × _{(5d )}0 e (5t2₎0 _{f (6x)}0+ ₂0 _{g 2m}0+ _(2m)0 _{h 2w}0× _3p0 i 12u0÷ _{3 j 3}2× ₅0 _{k (5a)}0+ _{4 l 8b}0− _(3b)0 m 6h0− _(6h)0 _n −_7c + _4c0 _{o (3e}2₎0− _(10e)0 _p q 1000− ₁₀₀₀0 _{r 3f}0+ ₄ − _{(5f )}0 _{s 36q}5÷ _12q5 _{t (3x}3₎3÷ _x9 u 60m5_n3÷ _12mn3 _{v 12p}0÷ _(2p)0 _{w (a}2_b3₎0 _{x 7} × _4k0 2 3 ---   0 5 4 ---   0 −1 2 ---   0 p q ---   0 ₃ 4 ---   0 1 2 ---   0 ₁ 2 --- y0 +

Exercise 7-05

Example 10 Example 11 Example 12

Working mathematically

Applying strategies and reasoning: Negative powers

1 a Copy and complete the following table of descending powers of 10. Use your

calculator if necessary. (Don’t be alarmed if the calculator gives decimal answers.) What rule did you use to complete the pattern?

b To see the hidden pattern clearly, you will need to change the decimals into fractions.

Copy and complete the following table. Express each decimal as a fraction, then write it as a power of 10. (The first two have been done for you.)

c Look carefully at the fractions written as powers of 10. What do you notice when you compare them with the corresponding negative powers of 10? Write down your findings. Write 10−7 _{and 10}−8 _{as fractions using the power of 10.}

d What does this tell you about negative powers?

e Write down what you have learnt about raising a number to a negative power.

Powers to ten 106 _{… 10}0 ₁₀−1 ₁₀−2 _{… 10}−6 Decimal form 1 000 000 … 0.1 0.01 … Powers of ten 10−1 10−2 … 10−6 Decimal form 0.1 0.01 … Fraction form … Fraction form

with powers of ten …

1 10 --- 1 100 ---1 101 --- 1 102 ---SkillBuilder 11-09 Exercising the four index laws

The negative index

Consider 24÷ ₂7=

= = =

2 a Copy and complete the table below. Use your calculator to express each power of 5 as

whole numbers or a fraction.

b 5−2 _{can be written as , or as . Use the table to write each of the following in two}

ways.

i 5−3 _{ii 5}−4 _{iii 5}−5

c Write each of the following in two ways.

i 4−2 _{ii 7}−3 _{iii 2}−6 3 Consider: = = = But = 104÷ ₁₀7 = 10−3 _{(Using Law 2)} So 10−3=

Using this method, simplify (in the two ways):

a to show that 2−5= _{b to show that 3}−1=

c to show that 5−6= _{d to show t h a t a}−2=

4 The reciprocal of 35 _is = ₃−5_.

Use negative indices to write the reciprocals of the following.

a 24 _{b 5}2 _{c 4 d k}5 _{e m}3

Compare your answers with those of other students.

53 ₅2 ₅1 ₅0 ₅−1 ₅−2 ₅−3 ₅−4 ₅−5 125 1 25 --- 1 52 ---= 1 25 --- 1 52 ---104 107 --- 10×10×10×10 10×10×10×10×10×10×10 ---1 10×10×10 ---1 103 ---104 107 ---1 103 ---23 28 --- 1 25 --- 3 4 35 --- 1 3 ---52 58 --- 1 56 --- a 4 a6 --- 1 a2 ---1 35 ---SkillBuilder 11-13 Division with a larger index in the denominator 24 27 ---2×2×2×2 2×2×2×2×2×2×2 ---1 1 1 1 1 1 1 1 1 2×2×2 ---1 23

---But 24÷ ₂7= ₂4−7 = 2−3 ∴ 2−3=

Proof:

a0÷ _an= = (since a0= ₁₎ But a0÷ _an= _a0 − n = a−n ∴ a−n= 1 23

---A negative power or index gives a fraction (with numerator 1):

a−m= 1 am ---a0 an ---1 an ---1 an

---Example 13

Express using positive indices:

a 3−1 _{b 4}−3 _{c k}−5

Solution

a 3−1= _{b 4}−3= _{c k}−5=

=

Express using positive indices:

a 3k−5 _{b a}2_b−3 _{c (5m)}−2

Solution

Evaluate 2−3_{, leaving your answer as a fraction.}

Solution

2−3= = = a 3k−5 = ₃ × _k−5 = × (3 = ) = b a2_b−3 = _a2 × b−3 = × = c (5m)−2 = = 1 31 --- 1 43 --- 1 k5 ---1 3

---Example 14

3 1 --- 1 k5 --- 3 1 ---3 k5 ---a2 1 --- 1 b3 ---a2 b3 ---1 5m ( )2 ---1 25m2

---Example 15

1 23 ---1 2×2×2 ---1

---Negative powers of quotients

Proof:

Consider = = 1 ÷ = 1 × = and = = = 1 ÷ = 1 × = = = = 1 ÷ = 1 × = and = = 1 ÷ = 1 × = = 2 3 ---   -1 ₁ 2 3 --- ---2 3 ---3 2 ---3 2 ---4 5 ---   -2 ₁ 4 5 ---   2 ---1 16 25 --- ---16 25 ---25 16 ---52 42 ---5 4 ---   2 = and = a b ---   -1 _b a --- a b ---   -n _b a ---   n a b ---   -1 ₁ a b --- ---a b ---b a ---b a ---a b ---   -n ₁ a b ---   n ---an bn ---bn an ---bn an ---b a ---   n

Example 16

Simplify the following and evaluate if possible.

a b c

Solution

a = = 1 b = = = 2 c = = = 4 5 ---   -1 ₃ 5 ---   -2 _2a b2 ---   -3 4 5 ---   -1 ₅ 4 ---1 4 ---3 5 ---   -2 ₅ 3 ---   2 25 9 ---7 9 ---2a b2 ---   -3 _b2 2a ---   3 b2 ( )3 2a ( )3 ---b6 8a3

---1 Express using positive indices:

a 5−2 _{b 3}−7 _{c 4}−1 _{d 8}−2

e 10−4 _{f m}−1 _{g h}−3 _{h w}−2

i 20−4 _{j (}−₁₁₎−1 _{k k}−8 _{l c}−6

2 Express using positive indices:

a 4d−1 _{b 3x}−5 _{c 2d}−3 _{d 4m}−2 _{e ab}−2

f m2_n−4 _{g wy}−2 _{h 4ac}−1 _{i 3p}−2 _{j 15kw}−4

k 12y2_m−3 _{l a}−4_m2 _{m d}−3_y3 _{n 4xy}−3 _{o v}−1_m−2 3 Write each of the following using positive indices.

a (2m)−1 _{b (xy)}−1 _{c (4h)}−2 _{d (5k)}−3

e (3h)−2 _{f (4k)}−3 _{g (2c)}−4 _{h (8y)}−1

4 Evaluate the following, leaving your answers as fractions.

a 3−2 _{b 4}−3 _{c 6}−1 _{d 7}−2

e 11−1 _{f 2}−5 _{g 4}−2 _{h 10}−2

5 Express using negative indices:

a b c d e f g h i j k l m n o p q r s t 6 Evaluate: a b c d e f g h

7 Simplify the following and evaluate if possible.

a b c d

e f g h

i j k l

8 Simplify each of the following, using positive indices.

a y5× _y−2 _{b e}−3× _e7 _c m × _m−1 _{d n}6× _n−5 _{e 4g}3× _3g−1 f 5a−2× _6a3 _{g 5x}−2× _{2x h 30e}−3× _2e−1 _{i 8p}−1÷ _2p2 _{j 8q} ÷ _2q−2 k 2r2÷ _8r−1 _{l 2t}−2÷ _8t−1 _{m (h}−1₎4 _{n (b)}−3 _{o (5x}−1₎2 1 m ---- 1 w ---- 1 8 --- 1 9 --- 1 22 ---1 n4 --- 1 34 --- 1 10-3 --- 1 e3 --- 1 t2 ----2 a --- 4 t2 ---- 2 w5 --- 5 d --- 1 2y ---1 7e --- 1 3a2 --- 5 3m4 --- 1 8 p3 --- 2 3k6 ---1 3 ---   -1 ₁ 4 ---   -2 ₂ 3 ---   -2 ₂ 5 ---   -3 2 3 ---   -1 ₃ 4 ---   -1 ₁ 10 ---   -5 ₅ 4 ---   -1 4 w ----   -1 _m n ----   -1 ₁ 4 ---   -1 ₄ 5 ---   -1 k 3 ---   -1 _x 3 ---   -2 _a2 4 ---   -3 4 3 ---   -2 − 2d 5 ---   -2 h 2 m3 ---     5 − a2c3 4 ---   -3 5d2 p3 ---     -3

Exercise 7-06

Example 13 Example 14 Example 15 Example 16 CAS 7-04 Negative indices SkillBuilder 11-14

The fifth index law

9 Simplify the following and express your answers in positive index form. a x3_y4× _x−3_y−5 _{b p}−4_q−1× _5p2_q−3 _{c (m}2_n3₎−2 d w3_p5÷ _w5_p3 _{e m}2_n3÷ _m−5_n−1 _{f 4a}3_bc2×−_2a−5_b−3_c−2 g 8xy3÷ _4x2_y7 _{h (6m}4₎−2× _9m−3 _{i p}2q × _p−3_q−1÷ _p4_q3 j 8a3_h−1÷−_4ah ÷ _a2_h3 _{k (a}2_k2₎−3× _(a−1_k2₎−2 _{l 4x}−3_y−1÷ _8xy3× _5x−1 m 4r4_t−3× _5r−5_t4 _{n -15ab}−2÷ _5a−1_b−3÷−_6ab7 _{o (d}−3_h−1₎−1÷−_4d3_h2 p (2v3_w−2₎5÷ _8v2_w−7 _{q 81a}−3_e−4÷ _(3a2_e−1₎4 _{r (c}−1_d−3₎3÷ _(c−1_d2₎−4 10 Evaluate the following, leaving your answers as fractions.

a 23× ₂−4 _{b (3}2₎−3÷ _{( 3}−3₎3 _{c 5}−1÷ ₂−1 d 3−2÷ ₂−1× _{6 e (4}−2₎2÷ _{( 2}−2₎3 _{f 3}2× ₍₃−2₎2

Squaring a number ending in 5, 1 or 9

Squaring a number ending in 5

The square of a number ending in 5 always ends in 25. For example, 352= _{325, and}

1052= _{11 025.}

A simple calculation trick requires three steps:

Step 1: Delete the 5 from the number.

Step 2: Multiply the remaining number by the next consecutive number. Step 3: Write ‘25’ at the end of the product.

1 Examine these examples: a 352

Deleting the 5 from 35 leaves just 3.

Multiplying 3 by the next consecutive number: 3 × 4 = 12 Writing ‘25’ at the end: 1225

352= ₁₂₂₅ b 1052

Deleting the 5 from 105 leaves 10. 10 × 11 = 110

11 025 1052= _{11 025} 2 Now calculate these:

a 252 _{b 55}2 _{c 45}2 _{d 85}2

e 1152 _{f 7.5}2 _{g 95}2 _{h 195}2

i 1.52 _{j 65}2 _{k 155}2 _{l 245}2

Squaring a number ending in 1

The square of a number ending in 1 always ends in 1. For example, 412= _{1681, and}

712= _5041.

A simple calculation trick requires three steps:

Step 1: Subtract 1 to round down to the nearest 10 and make a new number. Step 2: Square the new number.

Step 3: Add the new number and the next consecutive number to the square.

Skillbank 7

SkillTest 7-01

Squaring a number ending

3 Examine these examples: a 412 Round down to 40. Squaring 40: 402= ₁₆₀₀ Adding 40 and 41 to 1600: 1600 + 40 + 41 = 1681 412= ₁₆₈₁ b 712 702= ₄₉₀₀ 4900 + 70 + 71 = 5041 712= ₅₀₄₁

4 Now calculate these:

a 212 _{b 101}2 _{c 31}2 _{d 91}2

e 5.12 _{f 61}2 _{g 201}2 _{h 1.1}2

Squaring a number ending in 9

The square of a number ending in 9 also ends in 1. For example, 292= _{841, and 99}2= _9801.

A simple calculation trick requires three steps:

Step 1: Add 1 to round up to the nearest 10 and make a new number. Step 2: Square the new number.

Step 3: Subtract the new number and the previous consecutive number from the square.

5 Examine these examples: a 292

Rounding up gives 30. Squaring 30: 302= ₉₀₀

Subtracting 30 and 29 from 900: 900 − 30 − 29 = 841 292= ₈₄₁

b 992

1002= _{10 000}

10 000 − 100 − 99 = 9801 992= ₉₈₀₁

6 Now calculate these:

a 592 _{b 69}2 _{c 89}2 _{d 19}2

e 1092 _{f 4.9}2 _{g 79}2 _{h 11.9}2

Note: By combining and adapting the methods for squaring numbers ending in 5, 1 and 9, it

is also possible to square a number ending in 4 or 6.

Bonus trick: Squaring a two-digit number beginning with 1

This calculation trick requires three steps:

Step 1: Double the units digit and add 10. Step 2: Multiply by 10.

Step 3: Add the square of the units digit.

7 Examine these examples: a 172

Doubling the units digit and adding 10: 2 × 7 + 10 = 24 Multiplying by 10: 24 × 10 = 240

Adding the square of the units digit: 240 + 72= ₂₄₀ + ₄₉ = ₂₈₉

b 142

2 × 4 + 10 = 18 18 × 10 = 180

180 + 42= ₁₈₀ + ₁₆ = ₁₉₆

142= ₁₉₆ 8 Now calculate these:

a 122 _{b 13}2 _{c 18}2

d 192 _{e 11}2 _{f 1.6}2

Working mathematically

Reasoning and reflecting: Fractions as powers

(Spreadsheet optional)

1 Copy and complete this table of square numbers and their square roots.

2 Use your calculator to evaluate . (25 1 2 )

Now evaluate:

a b c d

3 Look at your calculator answers. Compare them with your answers to Question 1.

Write down what you notice. Predict the values of and .

4 What have you learnt about the fractional power ?

5 Repeat this investigation for fractional cube numbers and their cube roots. Copy and

complete this table.

6 Use your calculator to evaluate . (8 1 3 )

Square number Square root

1 1

4 2

9 3

100 10

Cube number Cube root

1 1 8 2 27 3

1000 10 25 1 2

---^

(

ab c ---

)

=

36 1 2 ---64 1 2 ---81 1 2 ---100 1 2 ---12 1 2 ---144 1 2 ---1 2 ---8 1 3

---^

(

ab c ---

)

=

The fractional index

The square root of a number is the value that, when squared, gives the number. For example,

= 8 because 82= ₈ × ₈ = _64.

The cube root of a number is the value that, when cubed, gives the number. For example,

= 5 because 53= ₅ × ₅ × ₅ = _125.

Now evaluate:

a b c d

7 What do you notice about your answers to Questions 5 and 6? 8 Explain what the fractional power means.

27 1 3 ---64 1 3 ---512 1 3 ---1000 1 3 ---1 3

---Calculating square roots

For computer spreadsheets

Step 1: Set up your spreadsheet as shown.

Step 2: Copy the formulas in cells C7, D6 and E6 to row 34. Step 3: Print your spreadsheet and paste it in your workbook.

1 Use your spreadsheet to find:

a b c d

2 Use your spreadsheet to find:

a b c d

3 a Compare your answers to Questions 1 and 2. Write what you notice. b Predict the values of and .

4 • In cell F5, insert the heading Number to the power 1/3. • In cell F6, enter the formula =C6^(1/3).

• Copy cell F6 down to row 34.

5 Suggest a meaning for:

a b

C D E

5 Number Square root of a number Number to power of 1/2

6 1 =SQRT(C6) =C6^(1/2) 7 =C6+1 8
34 9 13 25 29 9 1 2 ---13 1 2 ---25 1 2 ---29 1 2 ---36 1 2 ---64 1 2 ---n 1 2 ---n 1 3

---Using technology

Spreadsheet 7-02 Calculating square roots SkillBuilder 11-15 Simplifying fractions ( ) 64 3 ( ) 125 3

Consider × = = 41 = 4 But × = 2 × 2 = 4 ∴ =

Proof:

× = = a1 = a But × = a ∴ =

We also have that × × =

= 81 = 8 But × × = 2 × 2 × 2 ( = 2 because 2 × 2 × 2 = 8) = 8 ∴ =

Proof:

× × = = a1 = a But × × = a ∴ = 42 42 42 2 4 4 4 1 2 ---4

Any number raised to the power of is the square root of that number:

= 1 2 ---a 1 2 ---a a 1 2 ---a 1 2 ---a 1 2 --- 1 2 ---+ a a a 1 2 ---a 8 1 3 ---8 1 3 ---8 1 3 ---8 1 3 --- 1 3 --- 1 3 ---+ + 8 3 3 ₈ 3 ₈ 3 ₈ 8 1 3 ---8 3

Any number raised to the power of is the cube root of that number:

= 1 3 ---a 1 3 ---a 3 a 1 3 ---a 1 3 ---a 1 3 ---a 1 3 --- 1 3 --- 1 3 ---+ + a 3 3 _a 3 _a a 1 3 ---a 3

Example 17

Write each of the following with a fractional index.

Solution

a = b =

Write with a fractional index:

a b

Solution

a = b = Evaluate: a b

Solution

a = = 20 (because 202= ₄₀₀₎ (Calculator steps: 400 ) b = = 5 (because 53= ₁₂₅₎ (Calculator steps: 125 ) 5 5 1 2 ---11 3 ₁₁ 1 3

---Example 18

g 3 k g g 1 2 ---k 3 _k 1 3

---Example 19

400 1 2 ---125 1 3 ---400 1 2 ---400

√

–

=

125 1 3 ---125 3

√

–

3

=

1 Write the following, using fractional indices.

a b c d

e f g h

2 Write the following, using fractional indices.

a b c d

e f g h

3 Evaluate the following.

a b c d

e f g (−64) h

i (−8) j (−729) k l

4 Write the following, using either or .

a b c d

e f g h

5 Evaluate, correct to 2 decimal places:

a b c d 8 35 310 315 20 3 512 3100 72 m 3 w 38k ab 9y 3 _{xy 18 f} 3_10mn 64 1 2 ---343 1 3 ---1000 1 3 ---625 1 2 ---0.04 1 2 ---0.125 1 3 --- 1 3 ---1024 1 2 ---1 3 --- 1 3 ---196 1 2 ---900 1 2 ---3 37 1 2 ---8 1 3 ---d 1 2 ---20 1 2 ---4 p ( ) 1 3 ---100h ( ) 1 3 ---3c7 ( ) 1 2 ---w 1 3 ---4 1 3 ---8 1 2 ---100 3 ₁₀₀₀

Exercise 7-07

Example 17 Example 18 Example 19 CAS 7-05 Fractional indices