2. Dear children,
One more step in the journey towards
knowledge.
A book to learn math and learn it right
New thoughts, new deeds
New meaning of old ideas
Recognising the joy of learning
And the power of action
Moving yet forward.......
With regards,
Reshma
3. Exponents
The exponent of a number says how many times to use the number in a
multiplication.
In 82 the "2" says to use 8 twice in a multiplication,
so 82 = 8 × 8 = 64
In words: 82 could be called "8 to the power 2" or "8 to the second power",
or simply "8 squared"
Exponents are also called Powers or Indices.
Some more examples:
Example: 53 = 5 × 5 × 5 = 125
In words: 53 could be called "5 to the third power", "5 to the power
3" or simply "5 cubed"
Example: 24 = 2 × 2 × 2 × 2 = 16
In words: 24 could be called "2 to the fourth power" or "2 to the
power 4" or simply "2 to the 4th"
4. Exponents make it easier to write and use many multiplications
Example: 96 is easier to write and read than 9 × 9 × 9 × 9 × 9 × 9
You can multiply any number by itself as many times as you want
using exponents.
In general
an tells you to multiply a by itself,
so there are n of those a's:
Other Way of Writing It
Sometimes people use the ^ symbol, as it is easy to type.
Example: 2^4 is the same as 24
2^4 = 2 × 2 × 2 × 2 = 16
Negative Exponents
Negative? What could be the opposite of multiplying?
Dividing!
A negative exponent means how many times to divide one by the number.
Example: 8-1 = 1 ÷ 8 = 0.125
You can have many divides:
Example: 5-3 = 1 ÷ 5 ÷ 5 ÷ 5 = 0.008
But that can be done an easier way:
5. 5-3 could also be calculated like:
1 ÷ (5 × 5 × 5) = 1/53 = 1/125 = 0.008
In General
That last example showed an easier way to handle negative
exponents:
Calculate the positive exponent (an)
Then take the Reciprocal (i.e. 1/an)
More Examples:
Negative Exponent Reciprocal of Positive Exponent Answer
4-2 = 1 / 42 = 1/16 = 0.0625
10-3 = 1 / 103 = 1/1,000 = 0.001
(-2)-3 = 1 / (-2)3 = 1/(-8) = -0.125
What if the Exponent is 1, or 0?
1
If the exponent is 1, then you just have the number itself
(example 91 = 9)
0 If the exponent is 0, then you get 1 (example 90 = 1)
But what about 00 ? It could be either 1 or 0, and so people say it
is "indeterminate".
It All Makes Sense
6. My favorite method is to start with "1" and then multiply or divide as many
times as the exponent says, then you will get the right answer, for example:
Example: Powers of 5
52 1 × 5 × 5 25
51 1 × 5 5
50 1 1
5-1 1 ÷ 5 0.2
5-2 1 ÷ 5 ÷ 5 0.04
If you look at that table, you will see that positive, zero or negative
exponents are really part of the same (fairly simple) pattern.
Be Careful About Grouping
To avoid confusion, use parentheses () in cases like this:
With ( ) : (-2)2 = (-2) × (-2) = 4
Without ( ) : -22 = -(22) = - (2 × 2) = -4
With ( ) : (ab)2 = ab × ab
Without ( ) : ab2 = a × (b)2 = a × b × b
Let see this video
7. http://www.mathplanet.com/education/pre-algebra/discover-fractions-and-factors/powers-and-exponents
Let us solve some examples using the laws
of exponents.
Example 1
Find the value of
1. 2-3
2. ퟏ
ퟑ−ퟐ
Solution
1. 2-3 =
ퟏ
ퟐퟑ =
ퟏ
ퟖ
2.
ퟏ
ퟑ−ퟐ
= 32 = 3×3 = 9
Example 2
Simplify
1. (-4)5 ×(-4)10
2. 25 ÷ 2-6
Solution
1. (-4)5 ×(-4)10 = (-4)(5 -10) =(-4)(-5) =
1
(−4)5
2. 25 ÷ 2-6 = 2(5 -(-6)) = 211
Example 3
Express 4(-3) as a power with the base 2
Solution
We have 4 = 2 × 2 = 22
Therefore 4(-3) = (2×2)(-3) = (22)(-3) = 2(-6)
8. Use of exponents to express small numbers
in standard form
Observe the following facts
1. The distance from the earth to the sun is 149600000000 m.
2. The speed of light is 300000000 m/sec
3. Thickness of class vii mathematics book is 20mm
4. The average diameter of a red blood cell is 0.000007 mm
5. The thickness of human hair is in the range of 0.005 cm to 0.01
mm.
6. The distance of moon from the earth is 3844673000 m
7. The size of a plant cell is 0.00001275 m
8. Average radius of the sun is 695000km
9. Mass of propellant in a space shuttle solid rocket booster is 503600
kg.
10.Thickness of a piece of paper is 0.0016 cm
11.Diameter of a wire on a computer chip is 0.000003 m
12.The height of mount Everest is 8848 m
Observe that there few numbers which we can read like 2 cm, 8848
m, 695000 km. There are some large numbers like 150000000000
m and some very small numbers like 0.000007 m.
Identify very large and very small numbers from the above facts
and write them in the adjacent table
Very large numbers Very small numbers
150000000000 m 0.000007 m
