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Powers and roots
5.1
This unit will help you to:
calculatewhole-numberpowersofnumbers;
estimatesquareroots;
writeanumberastheproductofitsprimefactors;
findthehighestcommonfactor(theHCF)oftwonumbers;
findthelowestcommonmultiple(theLCM)oftwonumbers.
1 Working with integer powers of numbers
This lesson will help you to calculate whole-number powers of numbers and use the power keys of
a calculator.
Theshortwaytowrite22222isas25,or‘2tothepower5’.
Thesmallnumber5iscalledtheindex.Anindexcanbenegativeaswellaspositive.
Forexample,
1
1
92599 5
81
Thecalculatorkeytofindpowersofnumberslookslikethis: x y .
Example 1Findthevalueof64.
Keyin 6 x y 4 5 Thedisplayshowstheanswer:
1296
Tomultiplytwonumbersinindexform,addtheindices,soaman5am1n.
Example Simplify3432.
343253412536
Todividetwonumbersinindexform,subtracttheindices,soaman5amn.
Example 3Simplify5352.
5352553255155
N5.1 Powers and roots | 1
Exercise 1
1 Write each expression in index form.
a 2  2  2
b 4  4  4  4  4
c 3  3  3  3  3  3  3  3
e 1 55
d (1)  (1)  (1)  (1)
f 1
6
2 Work out each value without using your calculator. Show your working.
a 26
b (3)5
c 44
d (2)7
e 120
f (1)17
g 42
h 53
3 Use your calculator to work out each value.
Where appropriate, give your answer correct to two decimal places.
a 74
b 56
c 113
d 39
e (2)10
f 1.56
g 31.83
h 1.785 1 10.34
a 25  23
b 34  3
c 102  102
d a5  a3
e 56  52
f 125  12
g 84  84
h b5  b2
4 Simplify these.
5 A palindromic number reads the same forwards and backwards.
Copy and complete this cross-number puzzle.
1
Across
Down
1 A square number
1 23 × 5
3 A square palindromic number
2 A multiple of 13
2
3
4 A cube number
4
6 Some numbers can be written as the sum of three square numbers, e.g.
35 5 52 1 32 1 12
Write each of these numbers as the sum of three square numbers.
a 19
b 41
c 50
d 65
e 75
f 94
| N5.1
Powers and roots
7
Rachelis3yearsolderthanhersisterHannah.
Thesumofthesquaresoftheiragesinyearsis317.
HowoldareRachelandHannah?
Extension problem
8
Findthetwosmallestwholenumberswherethedifferenceoftheirsquaresisacube,and
thedifferenceoftheircubesisasquare.
Points to remember
Thenumber2raisedtothepower4is24or2222.
Inthenumber24,4iscalledtheindexorpower,and24issaidtobewritten
inindex form.
Tomultiplynumbersinindexform,addtheindices,soam×an5am1n.
Todividenumbersinindexform,subtracttheindices,soaman5amn.
Whenanegativenumberisraisedtoanevenpower,theresultispositive;
whenanegativenumberisraisedtoanoddpower,theresultisnegative.
2 Estimating square roots
This lesson will help you to estimate square roots and to use the square and cube root keys of
a calculator.
__
√ n isthesquare
___
rootofn.Forexample,√8159.
Youcanfindpositivesquarerootsonacalculator.
___
81√
Example 1Tofind√81,press:
Thedisplayshowstheanswer:
9
Onsomecalculatorsyoupressthesquare-rootkeyfirst: √ 8 1 Somecalculatorshaveacube-rootkey 3√ .
___
6 4 3√ Example Tofind3√ 64,press:
Thedisplayshowstheanswer:
4
N5.1 Powers and roots | 3
You can estimate the positive square root of a number that is not a perfect square.
___
Example 3 Estimate the value of √70.
___
___
___
___
√70
must lie between √64
and √81,
so 8  √70
 9. ___
Since 70 is closer to 64 than to 81, we expect √70
to be closer to 8 than to 9.
An estimate is 8.4.
√64
√70
√81
8
9
You can estimate the value of a square root more accurately by using trial and improvement.
Example Solve a2 = 135.
Value of a
11
12
11.5
11.6
11.7
11.65
Value of a2
121
144
132.25
134.56
136.89
135.7225
too small
too big
too small
too small
too big
too big
a is between 11 and 12.
a is between 11.5 and 12.
a is between 11.6 and 12.
a is between 11.6 and 11.7.
a is between 11.6 and 11.65.
So a must lie on the number line between 11.60 and 11.65.
11.65
11.6
11.7
Numbers between 11.6 and 11.65 round down to 11.6 to 1 d.p.
Answer: a  11.6 to 1 d.p.
Exercise 2
1 Write each expression in index form.
a x2  9
b x2  49
c x2  144
d x2  1
2 Write the value of each of these expressions. Use a calculator to help you.
Where appropriate, give your answer correct to two decimal places.
a
√2
e
3
√ | N5.1
b
125 Powers and roots
f
___
10 ___
√1.5
3
√ ____
c
3
√ d
√ 0.04 g
√3
h
3
√ (–64)
(–1)
3 Estimatetheintegerthatisclosesttothepositivevalueofeachofthese.
a
b
√6
___
c
√45
____
d
√ 155
___
√70
4 Use your calculatortofindthepositivevalueofeachofthesquarerootsinquestion3.
Giveyouranswerscorrecttoonedecimalplace.
5 Asquarepatioistobepavedwithsquarepavingslabs.Only
wholeslabswillbeused.
Thepavingslabshavetobeboughtinpacksof50.Fifteen
packsofslabsareneededtomakesurethatthereareenough
slabs.
Howmanyslabsareused?
6
Solvetheseequationsbyusingtrialandimprovement.Makeatabletohelpyou.
Giveyouranswersto1d.p.
a a2595
7
b a25152
c a25415
Theareaofthissquarerugis15m2.
Usetrialandimprovementtofindthelengthof
asidecorrecttotwodecimalplaces.
Extension problem
8
Whatisthesmallestsquarenumberthatbeginswiththree8s?
Points to remember
___
__
√ n isthesquarerootofn,e.g.√8159.
____
3
3
√ nisthecuberootofn,e.g. √ 12555,
√
2753.
N5.1 Powers and roots | 5
3 Prime factor decomposition
This lesson will help you to:
write a number as the product of its prime factors;
find the highest common factor (the HCF) of two numbers;
find the lowest common multiple (the LCM) of two numbers.
Did you know that…?
TheGreekmathematicianEuclidprovedinabout
300BCEwhatiscalledtheFundamentalTheorem
ofArithmetic.Thisshowsusthateveryinteger
canbewrittenasaproductofprimefactorsina
uniqueway.
Youcanuseadivision or ladder method tofindtheprime factorsofanumber.
Example 1
3
2
2
2
3 75
5 25
5 5
1
Theprimefactorsof75are5535523.
Theprimefactorsof24are32225323.
Youcanalsouseatree methodtofindtheprime factorsofanumber.
24
8
4
2
1
48
Example 4
12
4
2
Theprimefactorsof48are222235243.
3
2
2
2
Youcanuseprimefactorstofindthehighest common factor(HCF)andthelowest common
multiple(LCM)oftwonumbers.
Example 3
Theprimefactorsof72are22223.
Theprimefactorsof60are5322.
ShowtheseonaVenndiagram.
TheoverlappingorcommonprimefactorsgivetheHCF:
2235223512
AlltheprimefactorsgivetheLCM:
2223355233255180
6 | N5.1
N4.1
Powers
andofroots
Properties
numbers
72
60
3
2
2
2
3
5
Exercise 3
1 The number 18 can be written as the product of prime factors.
18  2  3  3  2  32
Write each of these numbers as the product of prime factors.
a 12
b 15
c 21
d 24
e 27
f 54
2 a List all the factors of 50.
b Write 50 as the product of prime factors.
3 a List all the factors of 45.
b Write 45 as the product of prime factors.
4 Find a number bigger than 50 that has the same number of factors as 50.
5 Using the Venn diagrams below, work out the HCF and LCM of:
a 72 and 30
b 50 and 80
30
72
3
2
2
80
50
2
5
5
c 48 and 84
2
5
3
2
84
48
2
2
2
2
2
3
2
7
6 The prime factors of 120 are 2, 2, 2, 3 and 5. The prime factors of 75 are 3, 5 and 5.
Show these numbers on a Venn diagram.
Use the diagram to work out the HCF and LCM of 120 and 75.
7 The prime factors of 40 are 2, 2, 2 and 5.
The prime factors of 90 are 2, 3, 3 and 5.
Show these numbers on a Venn diagram.
Use the diagram to work out the HCF and LCM of 40 and 90.
8 a Which prime numbers are factors of both 42 and 54?
b What is the biggest number that is a factor of both 42 and 54?
c What is the smallest number that is a multiple of both 42 and 54?
N5.1 Powers and roots | 9 FindtheHCFandLCMof:
a 28and40
10
b 200and175
c 36and64
Afour-digitnumberisamultipleof21andamultipleof35.
Whatisthesmallestnumberthatitcouldbe?
Extension problems
11
a Whatisthesmallestnumberwithexactly3factors?
b Whatisthesmallestnumberwithexactly5factors?
c W
hatisthesmallestnumberwithexactly7factors?Exactly9factors?11factors?
13factors?
12 Sevenfriendsarehavinglunchatthesamecafé.
Thefirstoneeatsthereeveryday,thesecondevery
otherday,thethirdeverythirdday,thefourth
everyfourthday,thefiftheveryfifthday,thesixth
everysixthday,andtheseventhonceaweekon
thesameday.
Thenexttimetheyallmeetatthecafétheyare
planningtohavealunchparty.
Inhowmanydaysfromnowwillthelunch
partybe?
Points to remember
Writinganumberastheproductofitsprimefactorsiscalledtheprime
factor decompositionofthenumber.
Example: 2452223or233.
Thehighest common factor(HCF)ofapairofnumbersisthelargest
numberthatisafactorofeachnumber
Example: 8hasprimefactors222;12hasprimefactors223.
Thehighestcommonfactoris22.
Thelowest common multiple(LCM)ofapairofnumbersisthesmallest
numberthatisamultipleofeachnumber.
Example: 8hasprimefactors222;12hasprimefactors223.
Thelowestcommonmultipleof8and12is2223548.
8 | N5.1
Powers and roots
How well are you doing?
Can you:
workoutwhole-numberpowersofnumbers?
estimatesquareroots?
writeanumberastheproductofitsprimefactors?
findthehighestcommonfactor(theHCF)oftwonumbers?
findthelowestcommonmultiple(theLCM)oftwonumbers?
Properties of numbers (no calculator)
1 2006 level 6
a Putthesevaluesinorderofsizewiththesmallestfirst.
52323324
b Lookatthisinformation.
55is3125
Whatis57?
2 2001 Paper 1 Level 6
a Lookatthesenumbers.
16
25
34
43
52
61
Whichisthelargest?
Whichisequalto92?
b Whichtwoofthenumbersbelowarenotsquarenumbers?
24
25
26
27
28
N5.1 Powers and roots | 9
3 Workoutthevalueofeachexpression.
a 3735
b 2423
c
3425
3224
4 Lookattheseequations.
a 24532a
Whatisthevalueofa?
b 28572b
Whatisthevalueofb?
5 a Findthehighestcommonfactorof84and60.
b Findthelowestcommonmultipleof16and36.
Properties of numbers (no calculator)
6 Suzythinksofanumber.
Sheuseshercalculatortosquarethenumberandthenadds5.
Heransweris29.01.
WhatisSuzy’snumber?
7 Thethreenumbersmissingfromtheboxesaredifferentprimenumbersbigger
than3.
c×c×c=1045
Whatarethemissingprimenumbers?
10 | N5.1
Powers and roots