WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
UNIT 5
1 ST LEVEL
POWERS AND ROOTS
SUMMARY
1. POWERS: DEFINITION AND TERMS
1.1 SPECIAL RULES ABOUT POWERS
1.2 POWERS OF TEN
1.2. 1 APPLICATIONS OF POWERS OF 10: Scientific Notation
1.3 LAWS OF EXPONENTS
2. ROOTS: DEFINITION AND TERMS
2.1 PERFECT SQUARES
2.2 HOW TO CALCULATE SQUARE ROOTS
EXTENSION:
PROPERTIES OF SQUARE ROOT RADICALS
Webs for more practice:
http://www.answermath.com/roo.htm roots
and exponents http://www.answermath.com/exp.htm
- http://www.bbc.co.uk/schools/ks3bitesize/maths/number/powers_roots/revise2.shtml
- http://www.321know.com/exp.htm
- http://www.emathematics.net/potencia.php?a=&pot=1
- www.321know.com/grade8/exponents
TEST:
http://www.bbc.co.uk/apps/ifl/schools/ks3bitesize/maths/quizengine?quiz=powers_roots&t
emplateStyle=maths
- www.bbc.co.uk/schools/k3bitesize/maths/numbers/powers_test
NOTE: THERE ARE pdf files: with more activities, problems and test of evaluation.
- powersTEN1.pdf / - powersTEN2.pdf / - ROOTSMath Worksheets.pdf / - / / rootsWORKsheet076.pdf / - testuunit5POWERS- ROOTS.pdf
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WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
UNIT 5
1 ST LEVEL
POWERS AND ROOTS
“Each problem that I solved became a rule which served afterwards to solve other
problems." - René Descartes
42 = 16
16 = 4
How many apples are there in total? 4+4+4+4= 4 x 4 = 42 = 16
How many apples are per row? 4 apples = 16 =4
1. POWERS: DEFINITION AND TERMS
A building has 2 floors, on each floor there are 2 apartments and on each apartment there
are 2 windows. How many windows are there in total? 2 x 2 x 2 = 23 = 8 windows.
2 floors
2 apartments per floor
2 windows per apartment
23 is a POWER
Invent similar problems, solve and propose them to the class.
DEFINITION: A power is a product of identical factors (repeated multiplication of
the same number).
TERMS:
The exponent tells the number of times a number is multiplied by itself (3). It is also called
index or indices in plural. We place the exponent to the upper right of the number.
The base is the number we multiply in the power, the repeated factor (2).
In general: we read the base raised to the n th power: 23 = two raised to the third power or
two cubed.
More examples: 72 = seven squared, (7 al cuadrado)
84 = eight raised to the fourth power or to the power of four. (8 elevado a 4 o a la potencia
de 4)
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WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
Exercises to solve and propose similar ones to class:
1. Complete the table:
POWER
BASE
EXPONENT
READ
58
10 Raised to the
power of 4
2. Write in powers:
a) 3 x 3 x 3=
b) 5 x 5 =
c) 2 x 2 x 2 x 2 x 2 x 2=
3. Calculate the value of these powers: a) 42=
b) 53=
d) 10 x 10 x 10=
c) 203 =
1.1 SPECIAL RULES ABOUT POWERS
Could you solve the next powers?
1. a) 15= 1 x 1 x 1 x 1 x 1= 1
b) 13= 1 x 1 x 1 = 1
c) 135 = 1 1 raised to any power
=1
2. a) 61= 6
b) 41=4
3. a) 20 =1
b) 450 = 1
c) 2341= 234 If the exponent is 1 the base doesn’t change.
5
=1
5
(see next section division of powers with the same base)
c) 50 = 1 If the exponent is 0 the power is 1. Why?
In conclusion, the special rules about powers are:
1. The number 1 raised to any power is 1 15= 1
2. Any number to the first power is that number: 81 = 8
3. Any number to the zero power is 1 650 = 1
http://www.dadsworksheets.com/v1/Worksheets/Exponents/Simple_Exponents_V1.html
1.2 POWERS OF TEN
Look at the next powers of 10: 10 0 = 1
101 = 10
104 = 10 x10 x 10 x 10 =10000
102 = 10 x 10 =100
105 =
103 = 10 x 10 x10 = 1000
106 =
Can you continue?
How many zeros are in the number? Which is the exponent? 10 2 = 100
(exponent 2 two zeros)
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WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
A power of base 10 is equal to the unit followed of as many zeros as the exponent
indicates.
Hence in reverse: 1000 = 103 and 100 = 102 Just count the number of zeros
and write the exponent: 4 zeros exponent = 4
Continue with the next numbers: a) 100000 =
b) 10000 =
c) 10 =
1.2. 1 APPLICATIONS OF POWERS OF 10
1. We apply powers of ten to express a number in expanding or polynomial form:
For example: 2345 = 2 x 1000 + 3 x 100 + 4 x 10 + 5 =
= 2 x 103 + 3 x 102 + 4 x 10 + 5
Work out these questions:
1. Express these numbers in expanding form-using powers of 10
A) 23567
B) 109861
C) 700560
2. Calculate:
a) 3 x 104 =
e) 80 =
b) 18 x 102 =
f) 145 =
c) 9, 8 x 103 =
g) 871 =
d) 0,643 x 102 =
h) 22 =
j) 105 =
3. Find the exponent:
A) 10 -- = 10000000
B) 10 -- = 1000
C) 5 -- = 25
D) 6-- = 6x6x6x6
2. SCIENTIFIC NOTATATION OR STANDARD FORM
We use scientific notation to write long numbers in shortened form. It is
use a lot in science. We write multiples of 10 using powers of 10.
For example: 5 280 000 = 5,8 x 106
Notice that:
1. We represent the number as a number between 1 and 9 (only one digit). 5,8
2. We write a multiplication sign and represent the number’s value to the power of ten. 106,
so we count the number of positions that the decimal point must be moved to determine
which power of 10 we use.
How to express 0, 00098 in scientific notation?
9,8 x 10-4
With very samll decimal numbers less than 1, but very long, we count the number of
decimals to the left and express it in a negative power, which means division: (see division
of powers for more explanation)
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WORKING ON MATHS IN ENGLISH
10 -1 =
1
= 0,1
10
Isabel Leo de Blas
10- 2 =
1
= 0, 01
100
10 -3 =
1
= 0, 001
1000
http://www.tutorvista.com/math/powers-of-ten-notation
Can you match the two columns?
A) 8,2 x 10-3
I) 38,200
B) 5 x 102
II) 500
C) 3,82 x 104
III) 1,624 x 103
D) 1624
IV) 8,24 x 10 0
E) 0,312
V) 3,12 x 10 -1
F) 8,24
VI) 0,0082
Solve these problems and investigate:
1. One scientist says that the size of a bacterium is 2 x 10 -6 mm.
Which decimal number is it? A) 0, 00002
B) 0,0000002
C) 0, 000002
2. The light speed is 300 000 km/s. Which is the correct scientific notation? A) 3 x 10 4
B) 3 x 10 6
C) 3 x 10 5
1.3 LAWS OF EXPONENTS
FOR WORKING WITH EXPONENTS YOU SHOULD KNOW THESE RULES:
1. To multiply powers with the same base, add their exponents. For example:
22 x 23 x 24 = 2 2 +3 + 4 = 2 9
2
2
Don’t use it to multiply powers with different bases: 2 x 3 (2 x 3) 2+2
2. To divide powers with the same base, subtract their exponents. For example:
35
3 : 3 = 2 = 3 5 -2 = 3 3
3
5
2
Don’t use it to divide powers with different bases: 5 : 2 (5 : 2) 4-3
Now we can understand why 10 -2 in scientific notation =
1 10 0
= 0,01
=
10 2 10 2
4
3
3. To raise a power to another power, multiply the exponents.
2 3
For example: (2 ) = 2
2x3
=26
5
WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
Webs: www.answermath.com/exponentiation and http://www.321know.com/exp.htm
PAY ATTENTION:
1. The power of a product is the product of the powers: (4 x 5)2 = 42 x 52
4 2
3
2. The power of a division is the division of the powers: =
42
32
2
2
4 2 (2 )
24
3. We can convert powers to prime factors: 4 x 2 = 2 x 2 = 2 and
= 3 = 3 =
8
2
2
=21= 2
3
2
3
5
ORDER OF OPERATIONS
Remember: 1st Brackets, ( )
2nd Exponents,
bn
3rd Multiply and divide x and :
4th Add and Subtract
+ and -
Solved exercise 5 + 16 : (5 -3) 2= 50+ 16 : 22
= 50 + 16 : 4
= 50 + 4 = 54
Now, solve next exercises and invent others to show in class:
A. 5+9 - 9 (22 +7) =
37 2
D. 2 + =
3 3
B. 52 x 53 – (2 +
5
2
2
1
)=
2
C. (4 + 9) x 5 : (8 – 23)=
22 2
F) =
2
3
E. 7 x 7 + 5 +2 =
2. ROOTS: DEFINITION AND TERMS
The reverse operation of squaring a number is finding its square root.
22 = 4 =2
or
22
4 = The square root of 4 is 2. Which is the 16 ,
64 ,
49 ?
4
6
=2
WORKING ON MATHS IN ENGLISH
3
The symbol
Which is
3
27,
8
3
=
2
Isabel Leo de Blas
since 2 x 2 x 2 = 23 = 8 The cube root of the number 8 is 2.
64 ?
In conclusion: The square root of a number is another number that multiplied by itself is
that number (square power). The cube root of a number is another number that multiplied
by itself 3 times gives that number (cubed power).
2.1 PERFECT SQUARES
The square root of a whole number may not be a whole number. 7 a whole number
because 2 x 2 = 4 and 3 x 3 = 9, Since 7 is not a product of two whole numbers, its square
root is between 2 and 3. Using the calculator we get 7 = 2,645751311 which is a decimal
number.
A whole number is a perfect square if its square root is also a whole number. For
example 16 = 4
Is 1 a perfect square? 1 = 1 Yes , it is
Is 4 a perfect square?
4 = 2 Yes, it is
Can you continue with more perfect squares? To do that just multiply the numbers from 1
to 10 by itself: 1x1, 2x2, 3 x3, 4x4, 5x5 ….
THE PERFECT SQUARES FROM 1 TO 10 ARE: 1, 4, 16, 25, 36, 49, 64, 81 AND 100
Is 225 a perfect square?
225 = 15, so 15 x 15= 225.
Also we can do the prime factoring: 225 = 52 x 32 Yes , it is a perfect square
And 325? 325 = 52 x 13 No, it is not a perfect square because 13 has not an even
exponent although 5 has an even exponent = 2.
Help me to solve these problems and invent similar ones:
1. My living room is 25 m2. I need to know the size of one side to buy some furniture. How
can I do it?
25m2
one side?
25 = 5 m
2. Peter has a square painting of 100 cm2 . He is looking for a frame. How long will it be?
FRAME?
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WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
2.2 HOW TO CALCULATE SQUARE ROOTS
What is the
75 ? Since it is not a perfect square you can do three things:
1. Use a calculator:
75 = 8,660254038
2. Approximate the root: look for the perfect square closest to the number to make an
estimate guess: 75 is between 64 and 81 so 64 = 8 and 81 = 9. Then, as it is
closest to 9, we estimate about 8,7
3. Simplify a square root: by writing the number as a product of numbers:
75 = 25x3 = 25x 3 = 5 3
Solved problem:
In a square orchard we plant 256 trees in equal rows? How many trees will we have for
row?
256 = 16 trees
result will be
- Invent a problem which
354
??????????
For long numbers you can find the square root grouping digits two by two starting from the
right 74556 : 7 45 56 After along process you get 273 as a root and 17 as a remainder.
EXTENSION:
PROPERTIES OF SQUARE ROOT RADICALS
1.
81 = 9x 9 = 3 x 3 The square root of a product is the product of the square roots.
2.
64
64 8
=
= Can you express this in English words? Work in groups.
9
3
9
3.
4.
( 6)
2
= 6 We can eliminate
and 2 because they are reverse operations.
4 + 9 4 + 9 13 2 + 3 = 5
<Find more exercises to practise
on line and on some pdfs activities
at page 1.
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