IGCSE Maths - Powers_teaching

P.1 NOT FOR SALE

© Copyright by Vinci Mak

Vinci & his team Powers, Roots, Reciprocals & Standard forms

IGCSE Mathematics (Edexcel)

Powers, Roots, Reciprocals

& Standard forms (Teaching version)

Notes Content Page

(I) Basic powers and roots 2

(II) Higher powers and roots 3

(III) Fractional (rational) indices 4

(IV) Negative powers and reciprocals 4

(V) The laws of indices 5

(VI) Standard (index) form 5

(VII) Calculating with numbers in standard form 6

(VIII) Integrated Practice 7

P.2 NOT FOR SALE



(I) Basic powers and roots

1. Square: a number multiplied itself 2 times

Square of 5 is 25 52 = 25

2. Square root: reverse the process of taking square

Square root of 25 has positive and negative values

Positive square root: 5 Negative square root: -5

If x2 = 25, then

x = 5 x = -5

3. Cube: a number multiplied itself 3 times

Cube of 5 is 125 53 = 125

4. Cube root: reverse the process of taking cube

Cube root of 125 is 5 Cube root of -125 is -5

51253= 51253

−=−

No negative cube root for 125

5. First 15 squares

12 = 1 6

2 = 36 11

2 = 121

22 = 4 7

2 = 49 12

2 = 144

32 = 9 8

2 = 64 13

2 = 169

42 = 16 9

2 = 81 14

2 = 196

52 = 25 10

2 = 100 15

2 = 225

6. First 10 cubes

13 = 1 6

3 = 216

23 = 8 7

3 = 343

33 = 27 8

3 = 512

43 = 64 9

3 = 729

53 = 125 10

3 = 1000

P.3 NOT FOR SALE



Quick Practice:

1. Find x if x2 = 72. Give your answers to 3 s.f.

Ans: x = 8.49 or -8.49

2. Find y if y3 = 38. Give your answers to 4 s.f.

Ans: y = 3.36

(II) Higher powers and roots

Higher powers Higher roots

xn: n factors of x multiplied together nx

1

: nth root of x

e.g. 64 = 6666 ××× = 1296 e.g. 5

1

32 is the 5th root of 32 = 2

Make sure that you know how to use your calculator’s power and root keys.

Quick Practice:

Use your calculator to obtain the values of:

(a) 144 (b) 1.56 (c) 4

1

1045 (d) 6

1

125

Round your answers to 3 significant figures where appropriate.

P.4 NOT FOR SALE



(III) Fractional (rational) indices

3

2

8

Without using a calculator, write the following expressions as simply as possible.

1. 3

4

125 2. 4

3

81 3. 2

5

100 4. 4

3

16

Ans: 1. 625 2. 27 3. 100000 4. 8

(IV) Negative powers and reciprocals

=−n

xn

x

1, where 0≠x

Reciprocal of a whole number : 1 divided by that number

e.g. Reciprocal of 2 is 2

1, which can also be written as 12− .

Quick Practice:

Evaluate these expressions, giving your answers as exact fractions.

1. 220− 2. 24− 3. 2

3

49

100−

4.

3

2

64

27−

Ans: 1. 400

1 2.

16

1 3.

1000

343 4.

9

16

The top of the fraction tells you what power to apply – squaring in this case.

The bottom of the fraction tells you what root to apply – cube root in this case.

Reciprocal of 2

2

1.

12−

.

P.5 NOT FOR SALE



(V) The laws of indices

Another name for a power is an index, so powers are often called indices.

There are several laws of indices that can help you to simplify index problems.

=×ba

xx bax

+ When multiplying, add the indices

=÷ba

xx bax

− When dividing, subtract the indices

=bax )( ab

x When raising to a power, multiply the indices.

=1x x Anything to the power one is the number itself

=0

x 1 Anything to the power zero equals 1

00 Not defined

Quick Practice:

1. 43 1010 × 2. 68 44 ÷ 3. 40 )3( 4. 02011)5( 5. 32 )10(

Ans: 1. 107 2. 4

2 3. 1 4. 1 5. 10

6

(VI) Standard (index) form

nd 10×

Quick Practice:

1. Write these numbers in standard form.

(a) 4,000,000,000 (b) 36,000 (c) 14,300,000

101 <≤ d . integer

P.6 NOT FOR SALE



2. Write these numbers in standard form.

(a) 0.0006 (b) 0.000 000 25 (c) 0.000 000 000 001 8

(VII) Calculating with numbers in standard form

Quick Practice:

If 81055.3 ×=a and 910065.2 ×=b use your calculator to work out the values of

each of these expressions. Give your answer in standard form, correct to 3 significant

figures.

(a) ab (b) b

a (c) ba 2+

P.7 NOT FOR SALE



(VIII) Integrated Practice

1. Find the value of 553322 ××××× .

Ans: 30

2. Evaluate

(a) 23− (b) 2

1

36 (c) 3

2

27 (d) 4

3

81

16−

Ans: (a) 9

1 (b) 6 (c) 9 (d)

8

27

3. Work out

(a) 40 (b) 24− (c) 2

3

16

Ans: (a) 1 (b) 16

1 (c) 64

4. Work out the values of

(a) 32 )2( (b) 2)3( (c) 924×

Ans: (a) 26 = 64 (b) 3 (c) 12

5. (a) Write 84 000 000 in standard form.

(b) Work out: 12104

84000000

×. Give your answer in standard form.

Solution: (a) 7104.8 × (b) 5101.2 −×

P.8 NOT FOR SALE



6. (a) Work out the value of 53.

(b) Work out the value of

(i) )5.05.4( 22− . Write down all the figures on your calculator display.

(ii) Write your answer correct to 2 decimal places.

Ans: (a) 125 (b)(i) 4.472135955 (ii) 4.47

7. Calculate the value of 2

98

1014.6

1032.41098.5−

×

×+×

Give your answer in standard form correct to 3 significant figures.

Ans: 101001.8 ×

8. 420 000 carrot seeds weigh 1 gram

a) Write the number 420 000 in standard form.

b) Calculate the weight, in grams, of one carrot seed.

Give your answer in standard form, correct to 2 significant figures.

Ans: (a) 5102.4 × (b) 6104.2420000

1 −×=

9. A floppy disk (old computer storage device) can hold 1 440 000 bytes of data.

a) Write the number 1 440 000 in standard form.

A hard disk can hold 9104.2 × bytes of data.

b) Calculate the number of floppy disks needed to store the 9104.2 × bytes of data.

Ans: (a) 61044.1 × (b) 166767.16661044.1

104.26

9

≈=×

×

10. ba

aby

+=

2 , 8103×=a and 7102×=b

Find y. Give your answer in standard form correct to 2 significant figures.

Ans: 3103.4 ×±

P.9 NOT FOR SALE



11. IGCSE Maths (Edexcel) 2008Nov Paper 4H Q12 modified

(a) a, b and c are positive numbers such that 101 <≤ ab and 101 <≤ c mcba 10)10()10( 74

×=×××

(i) Write down the value of m.

(ii) Find an expression for c in terms of a and b.

(b) )105.4()102.3( qpN ×××= , where p and q are integers.

Express N in terms of p and q.

Give your answer in scientific notation.

Solution

(a) (i) m = 11

(ii) ab

(b) 11044.1 ++×

qp

12. IGCSE Maths (Edexcel) 2008Jun Paper 4H Q20 modified

(a) Evaluate 1212 109105 ×+× . Give your answer in scientific notation.

(b) Each of the numbers p, q and r is greater than 1 and less than 10.

nrqp 101010 1515×=×+× and p + q > 10

(i) Find the value of n.

(ii) Find an expression for r in terms of p and q.

Solution

(a) 121014× � 13104.1 ×

(b) (i) 16

(ii) 10

qp +

P.10 NOT FOR SALE



IGCSE Edexcel 4MA0 Sample Assessment Material Paper 3H Q5

(a) Simplify, leaving your answers in index form,

(i) 35 77 ×

.................................

(1)

(ii) 39 55 ÷

.................................

(1)

(b) Solve 849

22

22=

×

n

n = .................................

(2)

IGCSE Edexcel 4MA0 Sample Assessment Material Paper 4H Q12

Simplify

(a) 2

43

a

aa ×

...........................

(2)

(b) 6)( x

...........................

(1)

(c) )1(6

)1(3 2

+

+

x

x

...........................

(2)

(a)(i) 78

(ii) 56

(b) 9 + 4 – n = 8

n = 5

(a) a5

(b) x3

(c) 2

1+x

P.11 NOT FOR SALE



IGCSE Edexcel 2011 Jun 4MA0 Paper 3H Q20 n

ax 10×= where n is an integer and 1010 <≤ a

Find, in standard form, an expression for x2.

Give your expression as simply as possible.

……………………

IGCSE Edexcel 2011 Jun 4MA0 Paper 4H Q13

The table gives the diameters, in metres, of four planets.

Planet Diameter (metres)

Mercury 61088.4 ×

Venus 71021.1 ×

Earth 71028.1 ×

Mars 61079.6 ×

(a) Which planet has the largest diameter?

.............................................................. (1)

(b) Write 61079.6 × as an ordinary number.

.............................................................. (1)

(c) Calculate the difference, in metres, between the diameter of Venus and the

diameter of Mercury. Give your answer in standard form.

....................................................... metres

(2)

x2 = n

a22 10× .

= 122

1010

+×

na

(a) Earth

(b) 6,790,000

(c) 7.22 x 106

P.12 NOT FOR SALE



IGCSE Edexcel 2010 Nov 4400 Paper 3H Q20

(a) Write 16

1 as a power of 2.

......................................

(2)

(b) Write 2 as a power of 8.

......................................

(2)


(a) Work out )104()109( 68×××

Give your answer in standard form.

.....................................

(1)

(b) mx 107×= and ny 105×= , where m and n are integers.

(i) It is given that 12105.3 ×=xy

Show that m + n = 11

(ii) It is also given that 27104.1 ×=y

x

Find the value of m and the value of n.

m = ..............................

n = ...............................

(5)

(a) 4216

1 −=

(b) 2 = 3

1

8

(a) 3.6 x 1015

(b) (i) xy = 35 x 10m+n

= 3.5 x 10m+n+1

m+n+1 = 12

m+n = 11

(ii) m – n = 27

Solving

m = 19 and n = -8

P.13 NOT FOR SALE




(a) Find the value of 42

1

)9(

…………………(1)

(b) Express 205 as a power of 25.

…………………(2)

(c) Express 8 as a power of 2.

…………………(2)

IGCSE CIE 0580 2011 Jun Paper 22 Q17

(a) 328 832 xx ÷ ,

Answer(a) ………………… [2]

(b) 3

23

64

x.

Answer(b) ………………… [2]

(a) 81

(b) 2510

(c) 21.5

(a) 244 −x

(b) 16

2x

P.14 NOT FOR SALE




(a) 32 )3( x

Answer(a) ………………… [2]

(b) 3

2

6 )125( x

Answer(b) ………………… [2]

IGCSE CIE 0580 2010 Nov Paper 21 Q4

Without using calculator, write the following in order of size, smallest first.

9.0 3 9.0 29.0 39.0

Answer ………… < ………… < ………… < ………… [2]

IGCSE CIE 0580 2010 Nov Paper 22 Q16

Simplify

(a) 2

1

16

81

16

x ,

Answer(a) ………………… [2]

(b) 7

410

32

416

y

yy−

×.

Answer(b) ………………… [2]

(a) 27x9

(b) 25x4

(a) 8

9

4x

(b) 2y-1

P.15 NOT FOR SALE



IGCSE CIE 0580 2010 Nov Paper 23 Q13 667 101010 ×=×+× cba

Find c in terms of a and b. Give your answer in its simplest form.

Answer c = ………………… [2]


Simplify

(a)

75.04

16

p,

Answer(a) ………………… [2]

(b) 2332 23 −−÷ qq .

Answer(b) ………………… [2]


Write 228 482 −×× in the form n2 .

Answer ………………… [2]


Change 64 square metres into square millimetres.

Give your answer in standard form.

Answer ………………… mm2 [2]

10a + b

(a) 8

3p

(b) 1

8

9 −q

210

7104.6 ×

P.16 NOT FOR SALE



Notes

IGCSE Maths - Powers_teaching

Documents

Transcript of IGCSE Maths - Powers_teaching