Chapter 11 Exponents and Powers
11.1 INTRODUCTION
Do you know what the mass of earth is? It is
Can you read this number?
Mass of Uranus is 86,800,000,000,000,000,000,000,000 kg.
Which has greater mass, Earth or Uranus?
Distance between Sun and Saturn is 1,433,500,000,000 m and distance between Saturn and Uranus is
These very large numbers are difficult to read, understand and compare. To make these numbers easy to read, understand and compare, we use exponents. In this Chapter, we shall learn about exponents and also learn how to use them.
11.2 EXPONENTS
We can write large numbers in a shorter form using exponents.
Observe
The short notation
We can similarly express 1,000 as a power of 10 . Note that
Here again,
Similarly,
In both these examples, the base is 10 ; in case of
We have used numbers like
This can be written as
Try writing these numbers in the same way
In all the above given examples, we have seen numbers whose base is 10 . However the base can be any other number also. For example:
Some powers have special names. For example,
Can you tell what
So, we can say 125 is the third power of 5 .
What is the exponent and the base in
Similarly,
In
In the same way,
TRY THESE
Find five more such examples, where a number is expressed in exponential form. Also identify the base and the exponent in each case.
You can also extend this way of writing when the base is a negative integer.
What does
It is
Is
Instead of taking a fixed number let us take any integer
TRY THESE
Express:
(i) 729 as a power of 3
(ii) 128 as a power of 2
(iii) 343 as a power of 7
Example 1 Express 256 as a power 2.
Solution
We have
So we can say that
Example 2 Which one is greater
Solution We have,
Since
Example 3 Which one is greater
Solution
Clearly,
Example 4 Expand
Solution
Note that in the case of terms
On the other hand,
Thus,
Example 5 Express the following numbers as a product of powers of prime factors:
(i) 72
(ii) 432
(iii) 1000
(iv) 16000
Solution
(i)
Thus,
(ii)
or
(required form)
(iii)
or
Atul wants to solve this example in another way:
or
Is Atul’s method correct?
Example 6 Work out
Solution
(i) We have
In fact, you will realise that 1 raised to any power is 1 .
(ii)
(iii)
You may check that
and
(iv)
(v)
EXERCISE 11.1
1. Find the value of:
(i)
(ii)
(iii)
(iv)
2. Express the following in exponential form:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
3. Express each of the following numbers using exponential notation:
(i) 512
(ii) 343
(iii) 729
(iv) 3125
4. Identify the greater number, wherever possible, in each of the following?
(i)
(ii)
(iii)
(iv)
(v)
5. Express each of the following as product of powers of their prime factors:
(i) 648
(ii) 405
(iii) 540
(iv) 3,600
6. Simplify:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
7. Simplify:
(i)
(ii)
(iii)
(iv)
8. Compare the following numbers:
(i)
(ii)
11.3 LAWS OF EXPONENTS
11.3.1 Multiplying Powers with the Same Base
(i) Let us calculate
Note that the base in
(ii)
Again, note that the base is same and the sum of exponents, i.e., 4 and 3 , is 7
(iii)
(Note: the base is the same and the sum of the exponents is
Similarly, verify:
Can you write the appropriate number in the box.
From this we can generalise that for any non-zero integer
TRY THESE
Simplify and write in exponential form:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Caution!
Consider
Can you add the exponents? No! Do you see ‘why’? The base of
11.3.2 Dividing Powers with the Same Base
Let us simplify
Thus
(Note, in
Similarly,
or
Let
or
Now can you answer quickly?
For non-zero integers
In general, for any non-zero integer
where
TRY THESE
Simplify and write in exponential form: eg.,
(i)
(iii)
(v)
11.3.3 Taking Power of a Power
Consider the following
Simplify
Now,
Thus
Thus
Similarly
Can you tell what would
So
From this we can generalise for any non-zero integer ’
TRY THESE
Simplify and write the answer in exponential form:
(i)
(iii)
Example 7 Can you tell which one is greater
Solution
but
Therefore
11.3.4 Multiplying Powers with the Same Exponents
Can you simplify
Now,
Consider
Consider, also,
In general, for any non-zero integer
TRY THESE
Put into another form using
(i)
(iii)
(iv)
(v)
Example 8 Express the following terms in the exponential form:
(i)
(ii)
(iii)
Solution
(i)
(ii)
(iii)
11.3.5 Dividing Powers with the Same Exponents
Observe the following simplifications:
(i)
(ii)
From these examples we may generalise
and
TRY THESE
Put into another form using
(i)
(ii)
(iii)
(iv)
(v)
Example 9 Expand:
(i)
(ii)
Solution
(i)
(ii)
What is
Obeserve the following pattern:
You can guess the value of
You find that
If you start from
- Numbers with exponent zero
Can you tell what
by using laws of exponents
So
Can you tell what
And
Therefore
Similarly
And
Thus
So, we can say that any number (except 0 ) raised to the power (or exponent) 0 is 1 .
11.4 MISCELLANEOUS EXAMPLES USING THE LAWS OF EXPONENTS
Let us solve some examples using rules of exponents developed.
Example 10 Write exponential form for
Solution
We have,
But we know that
Therefore
Example 11 Simplify and write the answer in the exponential form.
(i)
(ii)
(iii)
(iv)
(v)
Solution
(i)
(ii)
(iii)
(iv)
(v)
Therefore
Example 12 Simplify:
(i)
(ii)
(iii)
Solution
(i) We have
(ii)
(ii)
Note: In most of the examples that we have taken in this Chapter, the base of a power was taken an integer. But all the results of the chapter apply equally well to a base which is a rational number.
EXERCISE 11.2
1. Using laws of exponents, simplify and write the answer in exponential form:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
2. Simplify and express each of the following in exponential form:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
3. Say true or false and justify your answer:
(i)
(ii)
(iii)
(iv)
4. Express each of the following as a product of prime factors only in exponential form:
(i)
(ii) 270
(iii)
(iv) 768
5. Simplify:
(i)
(ii)
(iii)
11.5 DECIMAL NUMBER SYSTEM
Let us look at the expansion of 47561, which we already know:
We can express it using powers of 10 in the exponent form:
Therefore,
(Note
Let us expand another number:
Notice how the exponents of 10 start from a maximum value of 5 and go on decreasing by 1 at a step from the left to the right upto 0 .
11.6 EXPRESSING LARGE NUMBERS IN THE STANDARD FORM
Let us now go back to the beginning of the chapter. We said that large numbers can be conveniently expressed using exponents. We have not as yet shown this. We shall do so now.
1. Sun is located
2. Number of stars in our Galaxy is
3. Mass of the Earth is
These numbers are not convenient to write and read. To make it convenient we use powers.
Observe the following:
TRY THESE
Expand by expressing powers of 10 in the exponential form:
(i) 172
(ii) 5,643
(iii) 56,439
(iv)
We have expressed all these numbers in the standard form. Any number can be expressed as a decimal number between 1.0 and 10.0 including 1.0 multiplied by a power of 10. Such a form of a number is called its standard form. Thus,
Note, 5,985 can also be expressed as
We are now ready to express the large numbers we came across at the beginning of the chapter in this form.
The, distance of Sun from the centre of our Galaxy i.e.,
Now, can you express
Count the number of zeros in it. It is 10 .
So,

Mass of the Earth
Do you agree with the fact, that the number when written in the standard form is much easier to read, understand and compare than when the number is written with 25 digits? Now,
Simply by comparing the powers of 10 in the above two, you can tell that the mass of Uranus is greater than that of the Earth.
The distance between Sun and Saturn is
Can you tell which of the three distances is smallest?
Example 13 Express the following numbers in the standard form:
(i) 5985.3
(ii) 65,950
(iii)
(iv)
Solution
(i)
(ii)
(iii)
(iv)
A point to remember is that one less than the digit count (number of digits) to the left of the decimal point in a given number is the exponent of 10 in the standard form. Thus, in
EXERCISE 11.3
1. Write the following numbers in the expanded forms:
279404, 3006194, 2806196, 120719, 20068
2. Find the number from each of the following expanded forms:
(a)
(b)
(c)
(d)
3. Express the following numbers in standard form:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
4. Express the number appearing in the following statements in standard form.
(a) The distance between Earth and Moon is
(b) Speed of light in vacuum is
(c) Diameter of the Earth is 1,27,56,000 m.
(d) Diameter of the Sun is
(e) In a galaxy there are on an average 100,000,000,000 stars.
(f) The universe is estimated to be about 12,000,000,000 years old.
(g) The distance of the Sun from the centre of the Milky Way Galaxy is estimated to be
(h)
(i) The earth has 1,353,000,000 cubic
(j) The population of India was about 1,027,000,000 in March, 2001.
WHAT HAVE WE DISCUSSED?
1. Very large numbers are difficult to read, understand, compare and operate upon. To make all these easier, we use exponents, converting many of the large numbers in a shorter form.
2. The following are exponential forms of some numbers?
Here, 10, 3 and 2 are the bases, whereas 4,5 and 7 are their respective exponents. We also say, 10,000 is the
3. Numbers in exponential form obey certain laws, which are:
For any non-zero integers
(a)
(b)
(c)
(d)
(e)
(f)
(g)