N5 National 5 Portfolio
EF1.1 β Surds, Indices and Scientific Notation
Section A - Revision
This section will help you revise previous learning which is required in this topic.
R1 I can write whole numbers as products of two factors with one of the
factors a perfect square (where possible).
1. Write each number as a product of factors where one of the factors is a
perfect square.
(a) 27 (b) 12 (c) 32
(d) 75 (e) 48 (f) 8
(g) 50 (h) 125 (i) 20
R2 I have revised how to use the four operations in applications involving
negative numbers.
1. Evaluate
(a) β3 + 5 (b) 7 β 9 (c) 2 Γ (β7)
(d) β1 β (β6) (e) β3 Γ (β2) (f) β10 Γ· 5
(g) β4 Γ 2 (h) β24 Γ· (β6) (i) β8 + (β2)
(j) β2 β 7 (k) 4 + (β2) (l) 8 β 15
2. Evaluate
(a) β3
4 + 2 (b) β
3
2 β 1 (c)
3
8 Γ 5
(d) 3
4 Γ· β2 (e)
3
2 + (β
5
4) (f)
3
8β (β
5
4)
(g) β3
2 +
5
4 (h) β
3
8 Γ
5
2 (i)
3
4 Γ· (β
5
2)
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Section B - Assessment Standard Section
This section will help you practise for your Assessment Standard Test for Surds and
indices (Expressions and Formulae 1.1)
Practice Assessment Standard Questions
1. Simplify
(a) β27 (b) β12 (c) β32
(d) β75 (e) β48 (f) β8
(g) β50 (h) β125 (i) β20
2. Simplify
(a) π₯4Γπ₯3
π₯5 (b)
π¦3Γπ¦6
π¦2 (c)
π4Γπ3
π6
(d) π‘5Γπ‘
π‘3 (e)
π3Γπ3
π (f)
π3Γπ2
πβ1
(g) π5Γπ3
πβ3 (h)
π 5Γπ β1
π 3 (i)
πβ2Γπ7
π3
3. Simplify
(a) 4π₯2 Γ 2π₯3 (b) 3π₯3 Γ 5π₯5 (c) 2π₯3 Γ 6π₯β1
(d) 5π₯2 Γ 3π₯1
2 (e) 3π₯2 Γ 7π₯1
3 (f) 8π₯3 Γ 2π₯1
2
(g) 4π₯2 Γ 3π₯β1
2 (h) 3π₯3 Γ 10π₯β1
3 (i) 9π₯2 Γ 3π₯β1
2
4. A satellite travels 3 β 6 Γ 105 miles in a day. A higher orbit satellite travel 12 times this distance each day. Calculate the distance the higher orbit satellite travels each day. Give your answer in scientific notation.
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Section C β Operational Skills Section
This section provides problems with the operational skills associated with Surds
and Indices
O1 I understand the difference between rational and irrational numbers and
I know what a surd is.
1. For each of the numbers below, which are rational, write as a fraction with
whole numbers in both the numerator and denominator.
(a) β3 (b) π (c) β9
(d) β1
4 (e) β16 (f)
π
4
(g) β5 (h) β3
16 (i) β
9
25
O2 I can simplify, add, subtract, multiply and divide surds.
1. Simplify (without a calculator and showing all working)
(a) β24 (b) β18 (c) β45
(d) β80 (e) β72 (f) β108
(g) β24 + β6 (h) β2 + β18 (i) β45 β β5
(j) β80 β 2β5 (k) β72 + 3β2 (l) β108 β 3β3
(m) β24 + β54 β β6 (n) β125 + β80 β β20 (o) β6 Γ β15
(p) β14 Γ β7 (q) β48 Γ· β3 (r) β15 Γ β10
(s) β40
β5 (t)
β150
β6 (u) β12 Γ β30
2. Solve the following for π₯.
(a) βπ₯ + β18 = 4β2 (b) βπ₯ + β27 = β48 (c) β9π₯ β β5 = β20
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O3 I can rationalise a surd denominator.
Rationalise the surd denominator and simplify where appropriate
(a) 1
β5 (b)
2
β3 (c)
5
β7
(d) 4
β10 (e)
3
β6 (f)
14
β7
(g) β6
β15 (h)
β8
β5 (i)
10β3
β2
O4 I can multiply out brackets which involve surds
Multiply out the brackets and simplify where appropriate
(a) β2(β3 + 1) (b) β5(β2 β β3) (c) β2(β2 + β7)
(d) ββ11(β2 + 1) (e) β2(β3 + β2) β β6 (f) β3(β3 β β12)
(g) ββ5(β3 + β5) + β15 (h) β12(β3 + 1) β 2β3 (i) ββ7(β7 + 2)
O5 I can use the rules of indices πππ Γ πππ = πππ(π+π), πππ Γ· πππ =π
ππ(πβπ)
and abbba xkkx )(
β 0 1a and
1n
na
a
applying them to my previous
learning.
1. Simplify
(a) π₯2 Γ π₯5 (b) π¦3 Γ π¦β2 (c) π3 Γ 5π2
(d) 6π3 Γ 3π5 (e) 5β3 Γ 2ββ1 (f) π₯6 Γ· π₯2
(g) π7
π5 (h) π₯2 Γ· π₯3 (i) 10π¦4 Γ· 5π¦2
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2. Simplify
(a) (π₯2)3 (b) (π¦β2)4 (c) (π§β2)β5
(d) (3π3)2 (e) (2πβ1)5 (f) (5π¦β2)3
3. Write with positive indices
(a) π¦β5 (b) πβ1 (c) 3π₯β4
(d) 1
π‘β3 (e)
5
πβ7 (f)
2
5πβ7
(g) πβ3
4 (h)
5πβ1
2 (i)
πβ2
7
4. Simplify
(a) π¦2Γπ¦5
π¦3 (b)
π¦3Γπ¦β2
π¦β6 (c)
π8
π2Γπ4
(d) π
πβ1Γπ3 (e)
πβ2Γπβ3
πβ6 (f)
5πβ3Γ4π3
2
(g) π2Γπβ5
πβ3Γπ4 (h)
π 5Γ4π β5
2π β3 (i)
8π3Γ4πβ6
6π2Γπβ2
O6 I know that ( )m
mnna a and can apply this knowledge in problems.
1. Simplify, leaving the final answer with fractional indices.
(a) βπ (b) βπ3
(c) 1
βπ4
(d) βπ₯35 (e) βπ₯73
(f) 1
βπ₯34
(g) βπ₯ Γ βπ₯23 (h) 3π Γ βπ
3 (i)
βπ
βπ3
(j) 4βπ3
3π (k) 4π Γ βπ23
(l) βπ34Γ βπ53
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2. (a) Given that π¦ = π₯1
2, find π¦ when π₯ = 16.
(b) Given that π¦ = π₯2
3, find π¦ when π₯ = 64.
(c) Given that π¦ = π₯1
4, find π¦ when π₯ = 81.
(d) Given that π¦ = π₯1
3, find π¦ when π₯ = 125.
(e) Given that π¦ = π₯3
5, find π¦ when π₯ = 32.
O7 I can multiply out brackets which involve fractional or negative indices
1. Multiply out the brackets and simplify where appropriate
(a) π₯(π₯2 β π₯β2) (b) π3(πβ2 + π3) (c) πβ3(π + πβ1)
(d) 5π₯1
2 (2βπ₯ + 3π₯3
2) (e) 4π2(2πβ1 + 3πβ2) (f) π1
2 (βπ + πβ1
2)
(g) π‘β2(3π‘β2 β π‘2) (h) 3π2(2π2 + 7πβ4) (i) π1
3 (π2
3 + πβ1
3)
O8 I can convert large and small numbers to and from scientific notation.
1. Write the following numbers in scientific notation
(a) 7 000 (b) 650 000 (c) 4 120 000
(d) 820 (e) 37 100 000 000 (f) 1 345 000
(g) 3 πππππππ (h) 91
2 πππππππ (i) 16 β 2 πππππππ
2. Change each of the following back into normal form
(a) 8 Γ 105 (b) 3 β 25 Γ 104 (c) 7 β 153 Γ 108
(d) 4 β 03 Γ 107 (e) 2 β 8 Γ 106 (f) 5 β 55 Γ 1010
(g) 1 β 34 Γ 102 (h) 8 β 714 Γ 105 (i) 2 β 304 Γ 109
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3. Write the following numbers in scientific notation
(a) 0 β 04 (b) 0 β 000 062 (c) 0 β 357
(d) 0 β 000 000 002 4 (e) 0 β 000 095 (f) 0 β 6
(g) 0 β 000 012 53 (h) 0 β 000 000 000 236 (i) 0 β 001 65
4. Change each of the following back into normal form
(a) 3 Γ 10β4 (b) 7 β 5 Γ 10β2 (c) 6 β 8 Γ 10β6
(d) 5 β 07 Γ 10β3 (e) 4 β 8 Γ 10β5 (f) 5 β 3 Γ 10β7
(g) 4 β 344 Γ 10β5 (h) 9 β 94 Γ 10β6 (i) 5 β 34 Γ 10β1
5. Write each of the following in scientific notation
(a) 0 β 000 6 (b) 65 (c) 3 910
(d) 0 β 000 002 3 (e) 858 000 (f) 0 β 000 000 55
6. Change each of the following back into normal form
(a) 8 β 3 Γ 10β3 (b) 3 β 5 Γ 105 (c) 7 β 13 Γ 10β6
(d) 4 β 873 Γ 108 (e) 2 β 4 Γ 10β5 (f) 6 β 55 Γ 107
O9 I can solve problems involving multiplication and division of numbers
expressed in scientific notation with and without a calculator.
1. Do the following calculations without using a calculator and give your answer
in normal form
(a) 5 Γ (4 β 3 Γ 106) (b) 6 Γ (2 β 93 Γ 10β3)
(c) 9 β 3 Γ (7 Γ 105) (d) (4 β 8 Γ 107) Γ· 4
(e) (6 β 2 Γ 105) Γ· 2 (f) (7.2 Γ 10β3) Γ· 8
(g) (5 Γ 102 ) Γ (3 Γ 106) (h) (2 β 5 Γ 10β2 ) Γ (5 Γ 10β4)
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2. Use your calculator to work out the following and give your answer in
scientific notation
(a) (5 Γ 106 ) Γ· (8 Γ 10β4)
(b) 4 β 4 Γ (3 β 7 Γ 10β3)
(c) 9 β 3 Γ (6 Γ 105)
(d) (2 β 8 Γ 1010) Γ (5 β 4 Γ 103)
(e) (6 β 2 Γ 105)Β³
3. Complete the following calculations. Give your answers in scientific notation.
(a) There are 3 β 156 Γ 107seconds in a solar year.
How many seconds are there in 12 solar years?
(b) The Lotto jackpot of Β£9 β 3 Γ 106 was shared equally among 3 winners.
How much did each receive?
(c) A carbon atom weighs 2 β 03 Γ 10β23 grams.
What do 500 carbon atoms weigh?
(d) The orbit of a planet around a star is circular.
The radius of the orbit is 4 β 96 Γ 107 kilometres.
Calculate the circumference of the orbit.
(e) Radio signals travel at a speed of 3 Γ 108 metres per second.
A radio signal from Earth to a space probe takes 8 hours.
What is the distance from Earth to the probe?
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(f) One atom of gold weighs 3 β 27 Γ 10β22 grams.
How many atoms will there be in one kilogram of gold?
Give your answer in scientific notation correct to 2 significant figures.
(g) A snail crawls 3 kilometres in 16 days.
What is the average speed of the snail in metres per second?
Give your answer in scientific notation correct to 2 significant figures.
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Section D - Reasoning Skills Section
This section provides problems with Reasoning Skills in the context of Surds and
Indices
1. The area of the semi circle shown
is 4π₯3.
Show that the radius of the semi
circle is given by π =π₯
32
π
2. A cuboid has dimensions as
shown .
Show that the volume of the
cuboid is 100 cubic metres.
3. A particle travels 3ππ2 metres in 12π2π seconds.
Calculate the particles average speed in metres per second.
4. (a) Evaluate (24)2 .
(b) Hence find π, when (24)π =1
256
5. Lauren writes down the following statement.
π1
3 (π2
3 β πβ1
3) = π β 1
Is the statement true?
Justify your answer with working.
10πβ4 metres
2π metres
5π3 metres
π΄ = 4π₯3
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6. A square of side π₯ centimetres has
a diagonal which is 8 centimetres
long.
Show that π₯ = 4β2 cm.
7. Samβs homework jotter has the following statement.
6
β3 = 2β3
Is the statement true?
Justify your answer with working.
8. A small rectangle is drawn completely enclosed in a larger rectangle as
show.
With the dimensions given, show that the shaded area is β10 square
units.
9. Show that β3Γβ12
β3+β12=
2β3
3 .
β15
β6 β8
β5
8 cm
π₯ cm
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Answers
Section A
R1
Q1 (a) 9 Γ 3 (b) 4 Γ 3 (c) 16 Γ 2 (d) 25 Γ 3 (e) 16 Γ 3
(f) 4 Γ 2 (g) 25 Γ 2 (h) 25 Γ 5 (i) 4 Γ 5
R2
Q1 (a) 2 (b) β2 (c) β14 (d) 5 (e) 6
(f) β2 (g) β8 (h) 4 (i) β10 (j) β9
(k) 2 (l) β7
Q2 (a) 5
4 (b) β
5
2 (c)
15
8 (d) β
3
8 (e)
1
4
(f) 13
8 (g) β
1
4 (h) β
15
16 (i) β
3
10
Section B
Practice Assessment Standard Questions
Q1 (a) 3β3 (b) 2β3 (c) 4β2 (d) 5β3 (e) 4β3
(f) 2β2 (g) 5β2 (h) 5β5 (i) 2β5
Q2 (a) π₯2 (b) π¦7 (c) π (d) π‘3 (e) π5
(f) π6 (g) π11 (h) π (i) π2
Q3 (a) 8π₯5 (b) 15π₯8 (c) 12π₯2 (d) 15π₯5
2 (e) 21π₯7
3
(f) 16π₯7
2 (g) 12π₯3
2 (h) 30π₯8
3 (i) 27π₯3
2
Q4 4 β 32 Γ 106
Section C
O1
Q1 (c), (d), (e) and (i) are rational
O2
Q1 (a) 2β6 (b) 3β2 (c) 3β5 (d) 4β5 (e) 6β2
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(f) 6β3 (g) 3β6 (h) 4β2 (i) 2β5 (j) 2β5
(k) 9β2 (l) 3β3 (m) 4β6 (n) 7β5 (o) 3β10
(p) 7β2 (q) 4 (r) 5β6 (s) 2β2 (t) 5
(u) 6β10
Q2 (a) π₯ = 2 (b) π₯ = 3 (c) π₯ = 5
O3
(a) β5
5 (b)
2β3
3 (c)
5β7
7 (d)
2β10
5 (e)
β6
2
(f) 2β7 (g) β10
5 (h)
2β10
5 (i) 5β6
O4
(a) β6 + β2 (b) β10 β β15 (c) 2 + β14 (d) ββ22 β β11
(e) 2 (f) β3 (g) β5 (h) 6 (i) β7 β 2β7
O5
Q1 (a) π₯7 (b) π¦ (c) 5π5 (d) 18π8
(e) 10β2 (f) π₯4 (g) π2 (h) π₯β1 (i) 2π¦2
Q2 (a) π₯6 (b) 1
π¦8 (c) π§10 (d) 9π6
(e) 32
π5 (f)
125
π¦6
Q3 (a) 1
π¦5 (b)
1
π (c)
3
π₯4 (d) π‘3
(e) 5π7 (f) 2π7
5 (g)
1
4π3 (h)
5
2π (i)
1
7π2
Q4 (a) π¦4 (b) π¦7 (c) π2 (d) 1
π
(e) π (f) 10 (g) 1
π4 (h) 2π 3 (i)
16
3π3
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O6
Q1 (a) π1
2 (b) π1
3 (c) 1
π1
4
or πβ1
4 (d) π₯3
5
(e) π₯7
3 (f) 1
π₯3
4
or π₯β3
4 (g) π₯7
6 (h) 3π4
3 (i) π1
6
(j) 4π
12
3 (k) 4π
5
3 (l) π29
12
Q2 (a) π¦ = 4 (b) π¦ = 16 (c) π¦ = 3 (d) π¦ = 5 (e) π¦ = 8
O7
Q1 (a) π₯Β³ β1
π₯ (b) π + π6 (c)
1
π2 +
1
π4 (d) 10π₯ + 15π₯2
(e) 8π + 12 (f) π + 1 (g) 3
π‘4 β 1 (h) 6π4 +21
π2 (i) π + 1
O8
Q1 (a) 7 Γ 103 (b) 6 β 5 Γ 105 (c) 4 β 12 Γ 106 (d) 8 β 2 Γ 102
(e) 3 β 71 Γ 1010 (f) 1 β 345 Γ 106 (g) 3 Γ 106 (h) 9 β 5 Γ 106
(i) 1 β 62 Γ 107
Q2 (a) 800 000 (b) 32 500 (c) 715 300 000 (d) 40 300 000
(e) 2 800 000 (f) 55 500 000 000 (g) 134
(h) 871 400 (i) 2 304 000 000
Q3 (a) 4 Γ 10β2 (b) 6 β 2 Γ 10β5 (c) 3 β 57 Γ 10β1 (d) 2 β 4 Γ 10β9
(e) 9 β 5 Γ 10β5 (f) 6 Γ 10β1 (g) 1 β 253 Γ 10β5 (h) 2 β 36 Γ 10β10
(i) 1 β 65 Γ 10β3
Q4 (a) 0 β 0003 (b) 0 β 075 (c) 0 β 0000068 (d) 0 β 00507
(e) 0 β 000048 (f) 0 β 00000053 (g) 0 β 00004344 (h) 0 β 00000994
(i) 0 β 534
Q5 (a) 6 Γ 10β4 (b) 6 β 5 Γ 101 (c) 3 β 91 Γ 103 (d) 2 β 3 Γ 10β6
(e) 8 β 58 Γ 105 (f) 5 β 5 Γ 10β7
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Q6 (a) 0 β 0083 (b) 350 000 (c) 0 β 00000713 (d) 487 300 000
(e) 0 β 000024 (f) 65 500 000
O9
Q1 (a) 2 β 15 Γ 107 (b) 1 β 758 Γ 10β2 (c) 6 β 51 Γ 106 (d) 1 β 2 Γ 107
(e) 3 β 1 Γ 105 (f) 9 Γ 10β4 (g) 1 β 5 Γ 109 (h) 1 β 25 Γ 10β5
Q2 (a) 6 β 25 Γ 109 (b) 1 β 628 Γ 10β2 (c) 5 β 58 Γ 106 (d) 1 β 512 Γ 1014
(e) 2 β 38328 Γ 1017
Q3 (a) 3 β 7872 Γ 108 (b) 3 β 1 Γ 106 (c) 1 β 015 Γ 10β20
(d) 3 β 1 Γ 108 (e) 8 β 64 Γ 1012 (f) 3 β 1 Γ 1024 (g) 2 β 2 Γ 10β3
Section D - Reasoning Skills Section
Q1 Proof Q2 Proof Q3 π2
4ππ Q4 π = β2
Q5 β 9 Proof.