N
Powers and roots
6.1
This unit will help you to:
use ‘trial and improvement’ to estimate square roots and cube roots;
multiply and divide positive and negative integer powers;
use facts that you know to work out new facts;
write numbers in standard form.
1 Squares, cubes and roots
This lesson will help you to estimate square roots and cube roots using ‘trial and improvement’.
Did you know that…?
The word in the English language with more Zs than any other word is
zenzizenzizenzic. It means the eighth power of a number, e.g. the zenzi-zenzizenzic of 2 is 28  256.
The word was suggested in 1557 in The Whetstone of Witte, by Robert
Recorde, a 16th-century Welsh writer of popular mathematics textbooks,
born in Tenby in 1510.
The German word zenzic, from the Italian censo, means ‘squared’. So the fourth
power of a number, which is the square of a square, is zenzi-zenzic, and the
eighth power is zenzi-zenzi-zenzic. Similarly, the sixth power is zenzi-cube,
or the square of a cube.
3√n is the cube root of n.
The cube root of a positive number is positive.
The cube root of a negative number is negative.
Some calculators have a cube root key 3√ .
Example 1
To find
3
To find
3
___
√64,
press:
6 4
√64,

64, press:
√
3
The display shows:
6 4  3√ The display shows:
4
-4
N6.1 Powers and roots | 1
You can estimate the cube root of a number that is not a perfect cube.
___
Example 2 Estimate the value of 3√ 70.
3
___
Since 70 lies between 43 and 53, 4  √ 70
 5.
3
___
Since 70 is closer to 64 than to 125, √ 70
is likely to be closer to 4 than to 5. An estimate is 4.2.
3
3
÷64
3
÷70
÷125
4
5
You can estimate the value of a root more accurately by using trial and improvement.
Example 3 Solve a3 = 240.
Value of a
6
7
6.2
6.3
6.22
6.21
6.215
Value of a3
216
343
238.328
250.047
240.641 848
239.483 061
240.061 988 4
too small
too big
too small
too big
too big
too small
too big
a is between 6 and 7
a is between 6.2 and 7
a is between 6.2 and 6.3
a is between 6.2 and 6.22
a is between 6.21 and 6.22
a is between 6.210 and 6.215
So a must lie on the number line between 6.21 and 6.215.
6.21
6.215
6.22
Numbers between 6.21 and 6.215 round down to 6.21 to 2 d.p.
Answer: a  6.21 to 2 d.p.
Exercise 1
1 Solve each equation.
a x3  64
b x3   27
c x3  125
d x3   1
2 Use a calculator to find the value of each expression correct to two decimal places.
a
| N6.1
√5
Powers and roots
b
____
100 3
√ c
3
√ (–80)
d
___
√0.9
3
Estimate the integer that is closest to the value of each of these.
a
3
___
√40
b
3
___
c
√85
3
____
d
√ 550
3
____
√ 900
4
Use a calculator to find the cube roots in question 3 correct to two decimal places.
5
Solve these equations by using trial and improvement.
Make a table to help you.
Give your answers correct to one decimal place.
a a3  14
6
b a3  7000
c a(a  2) 10
This Indian box is in the shape of a cube.
Its volume is 800 cm3.
Use trial and improvement to find the length
of a side correct to two decimal places.
7
Any positive whole number can be written as the sum of four square numbers.
For example,
23  1  4  9  9
Investigate different ways of writing 150 as the sum of four square numbers.
How many different ways can you find?
Can you write 150 as the sum of three square numbers?
Extension problem
8
I am an odd two-digit number but I am not a prime number.
If you reverse me and add me to myself, you get a square number.
If you reverse me and subtract me from myself, you get another square number.
Who am I?
Points to remember
__
√ n
is the square root of n. The square root can be positive or negative.
√n is the cube root of n. The cube root of a positive number is positive,
and of a negative number is negative.
You can estimate square roots and cube roots using trial and
improvement.
3
N6.1 Powers and roots | 3
2 Equivalent calculations using powers of 10
This lesson will help you to multiply and divide by powers of 10 and use known facts to work out
new facts.
Did you know that…?
A googol is 10 to the power 100, or 10100.
This is 1 with one hundred zeros.
A googolplex is 10 to the power googol.
The headquarters of Google, the Internet search
engine, at Mountain View, California, is called
Googleplex, named using a variation of googol. Sergey Brin and Larry Page, of Google
Multiplying the numerator by 10 multiplies the answer by 10.
Multiplying the denominator by 10 divides the answer by 10.
For example, 6.8  34 so 68  340 and 6.8  3.4
0.2
0.2
2
Dividing the numerator by 10 divides the answer by 10.
Dividing the denominator by 10 multiplies the answer by 10.
For example ,6.8  34 so 0.68  3.4 and 6.8  340
0.2
0.2
0.02
Example 1:
Given that 16.3  6.52, work out the value of 16.3
2.5
25
16.3  16.3  6.52  10  0.652
25
2.5  10
Multiplying the denominator by 10 divides the
answer by 10.
Example 2:
Given that 3.46  25.5  25.95, work out the value of 34.6  2.55.
3.4
0.34
34.6  2.55  3.46  25.5
0.34
3.4  10
 25.95  10
 259.5
4 | N6.1
N4.1
Powers
andofroots
Properties
numbers
Multiplying 3.46 by 10 to get 34.6, and dividing 25.5 by 10 to get
2.55, does not alter the numerator.
Dividing the denominator 3.4 by 10 to get 0.34 multiplies the
answer 25.95 by 10.
Exercise 2
Do these questions without using a calculator. Show your working.
1
Given that 5.5  6.6  36.3, work out:
a 5.5  66
2
7
c 0.183  3.75
d 0.183  37.5
b 1288  5.6
c 12.88  0.23
d 0.023  0.56
b 1.512  0.072
c 15.12  21
d 1.512  2.1
c 4.42  130
d 44.2  0.34
c 0.464  510
3.4
d 4.64  5.1
0.34
c 7.2  5.6
9.6
d 7200  560
9600
Given that 442  34  13, work out:
a 4.42  340
6
b 1.83  3.75
Given that 1512  72  21, work out:
a 15.12  7.2
5
d 550  0.066
Given that 23 × 56  1288, work out:
a 0.23  560
4
c 55  0.66
Given that 18.3  3.75  4.88, work out:
a 183  3.75
3
b 0.55  0.66
b 442  1.3
Given that 46.4  5.1  69.6, work out:
3.4
a 46.4  51
b 464  51
34
3400
Given that 72  56 = 96, work out:
42
a 7.2  56
b 0.72  5.6
0.42
420
Points to remember
Use facts that you know to work out new facts.
If you multiply the numerator or divide the denominator of a fraction by
a power of 10, the answer is multiplied by the same power of 10.
e.g. 3.8 = 19,
0.2
3.8  10 = 190,
0.2
3.8 = 190
0.2  10
If you divide the numerator or multiply the denominator of a fraction by
a power of 10, the answer is divided by the same power of 10.
e.g. 3.8 = 19,
0.2
3.8  10 = 1.9,
0.2
3.8 = 1.9
0.2  10
N6.1 Powers and roots | 5
3 Standard form
This lesson will help you to use the index laws and express numbers in standard form.
Did you know that…?
This puzzle is almost 4000 years old. It was
invented by the ancient Egyptians and recorded
by the scribe Ahmes in 1650 BCE on the Rhind
Papyrus.
A wealthy Egyptian farmer owned seven barns.
Each of the barns housed seven cats.
Each of the cats caught seven mice.
Each of the mice would have eaten seven sheaves
of wheat. Each sheaf of wheat produced seven
measures of flour.
How many measures of flour did the farmer’s cats save?
7  7  7  7  7  75  16 807
A similar puzzle exists today in the traditional English version:
‘As I was going to St Ives …’
10 000  10  10  10  10, which is 104 or ‘10 to the power 4’.
The small number 4 is called the index.
1
1
An index can be negative as well as positive. For example, 102  102  100.
To multiply two numbers in index form, add the indices, so 10m  10n  10mn.
For example, 104  102  104  2  106.
To divide two numbers in index form, subtract the indices, so 10m  10n  10mn.
For example, 103  102  1032  101  10.
To raise the power of a number to a power, multiply the indices, so (am)n  amn.
For example, (53)2  53  53  56  532.
Exercise 3A
1
Work out the value of each expression.
a 21
6 | N6.1
Powers and roots
b 32
c 103
d 20
2 Simplify these.
a 32  33
b 42  4
c 104  102
d 2  22  25
e 42  43
f 32  31
g 54  53
h 102  103
b (52)2
c (102)3
d (10)10
3 Simplify these.
a (22)4
4 Simplify these.
a
24
27  22
b
52  54 5
c
43  43 42
d
24  22
27  21
5 Find the value of n in each equation.
a 32  2n
b 2n  22  25
c 2n  43
d 100  22  5n
Exercise 3B
We sometimes need to write very large or very small numbers. For example, the age of the
Earth is about 4.6 thousand million years. In full, this is 4 600 000 000 years.
A number written in standard form has the form A  10n, where A is a number between
1 and 10 and n is an integer.
Example 1
Write 9000 in standard form.
9000  9  1000  9  103 in standard form.
9 is the number between 1 and 10, so A  9 and n  3.
Th
H
T
U
9
9
0
0
0
The 9 has moved three places to the left because it has been multiplied by 103.
Numbers in standard form can be written as ordinary numbers.
Example 2
Write 8  106 as an ordinary number.
Solution: 8  106  8 000 000
The 8 moves six places to the left from the units place to the millions place.
N6.1 Powers and roots | 1
2
3
Write each number in standard form.
a 58 000 000
b 0.000 37
c 225 000
d 49 300
e 0.0002
f 26 789
g 0.0043
g 0.000 000 15
Write each standard form number as an ordinary number.
a 8.6  104
b 4.21  103
c 7.8  103
d 3.25  102
e 7  109
f 4.13  104
g 6.9  106
g 2.01  101
c 0.33  103
d 28  104
Write each number in standard form.
a 26  103
4
b 47.2  105
Write in order these numbers in standard form, starting with the smallest.
1.6  104
3.7  101
4.6  103
1.9  102
2.3  102
Extension problems
5
For what values of n is 7n  3n a multiple of 10?
6
Show that (7  106)  (6  104)  4.2  1011.
7
Write each expression as a number in standard form.
a (4  108)  (2  103)
b (6  105)  (1.5  103)
c (4  102)  (3  105)
d (6  107)  (3 × 106)
e (6  109)  (5 × 103)
f (5  108)  (2 × 106)
Points to remember
To multiply two numbers in index form, add the indices,
so am  an  amn.
To divide two numbers in index form, subtract the indices,
so am ÷ an  amn.
To raise the power of a number to a power, multiply the indices,
so (am)n  amn.
A number in standard form is of the form A  10n,
where 1  A  10 and n is an integer.
8 | N6.1
Powers and roots
How well are you doing?
Can you:
calculate and estimate square roots and cube roots?
multiply and divide positive and negative integer powers?
use facts you know to work out new facts?
write numbers in standard form?
Powers and roots (no calculator)
1
Write the value of each expression correct to two decimal places.
a
2
___
b
√19
3
____
c
√ 400
3
√(–150)
( –150)
d
____
√ 0.08
1999 level 6
The length of one side of a rectangle is y.
This equation shows the area of the rectangle:
y(y  2)  67.89
Find the value of y. Show your working.
You may find the following table helpful.
3
y
y2
y(y  2)
8
10
80
too large
1999 level 7
a
Write the values of k and m.
64 = 82 = 4k = 2m
b Here is some information.
215 = 32 768
What is the value of 214?
N6.1 Powers and roots | 9
4
2005 level 7
Here is an equation.
xy = 64
Give four different pairs of values for x and y that satisfy this equation.
5
2007 level 7
Work out the values of m and n.
58  54 = 5m
58
n
4=5
5
6
2005 level 8
What is (4  108)  (8  104)?
Write your answer in standard form.
7
2007 level 8
One light-year is approximately 9 430 000 000 000 kilometres.
Write this distance in standard form.
10 | N6.1
Powers and roots