What are exponents?
A longer number which can be written its same factors e.g. if x is multiplied n times by itself, it would look
like
a × a × a × a × a × a × a × a ×…. n times, where a is
an integer. To short such numbers they can be written in exponent form like an.
Here, an is pronounced as a raised to power n or nth power of a.
x is called base and n is the power or the exponent.
2 is multiplied 5 times
It can be written as 2 × 2 × 2 × 2 × 2 × = 32
Its exponent form can be written as 25 and read as 2 raised to power 5 or 5th power of 2
In 25, 2 is the base and 5 is the power or exponent.
Base should not be zero in the exponent form.
Next are the rules to change any number into its exponent form, these rules are also known as Laws of Exponents.
Laws of exponents
There are six important laws of exponents those help to solve problems of exponents when their bases or powers are multiplied or divided.
1. When same bases are multiplied
am × an is such case where bases are same i.e. a, where a ≠ 0.
As per this rule, if same bases, which is a, are multiplied, the powers m and n will get added up and the
new common base will be again a
i.e. new power will be m + n and the base will be a, which can be written as am + n
or am × an = am + n
23 × 22
= 23 + 2
= 25
2. When same bases are divided
aman or
am ÷ an
is an example where same bases a are divided.
According this rule, if same bases are divided then powers m and n will be subtracted.
i.e. base a will have new power will m – n, which can be written as am – n
∴ aman
= am – n
2624
= 26 – 4
= 22
3. When exponent is 0
When exponent of a number is 0, then that becomes equal to only 1.
a0 = 1, where a ≠ 0
Example 1: 20
= 1
Example 2: 2424
= 24 – 4
= 20
= 1
4. Power of a power
When a power has its own power, then the powers get multiplied.
(am)n = am × n = amn, where a ≠ 0
(52)3
= 52 × 3
= 56
5. Product of bases with same exponents
When the numbers with different bases and same exponents are multiplied, then a common exponent is set on the
multiplication value of the bases.
am × bm = (a × b)m = (ab)m
35 × 25
= (3 × 2)5
= 65
6. Division of bases with same exponents
In the division of numbers with different bases and same exponents, then the two numbers are divided and a
common exponent is set on the division of the bases.
ambm
= (ab)m
4565
= (46)5
What is standard index form?
Standard Index Form or Standard Form (in United Kingdom) or Scientific Notation is a way of writing the long and short numbers using exponents or powers.
Long and small numbers can be written in the form of exponents to make them short and readable. Any number which is written in the form of k × 10n, where k is the natural or a decimal number with ones place only, is the standard index form.
Standard form of long number
625481453588462135 is a long number.
It can be written as 6.25 × 1017.
Standard form of small number
0.00000005 is an example of small number.
0.00000005 can be written as 5 ×10-8.
There are actual scientific constant values which are also written in standard forms are listed in tables below.
Table: list of examples of exponent forms of long numbers.
| Name | Standard Index Form |
|---|---|
| Mass of earth | 5.9722 × 1024 kg |
| Distance of sun from earth | 1.496 × 108Km |
| Speed of light | 3 × 108 m/s |
Table: list of examples of exponent forms of short numbers.
| Name | Standard Index Form |
|---|---|
| Mass of electron | 9.109 × 10-31 kg |
| Mass of proton | 1.672 × 10-27 kg |
| Mass of neutron | 1.674 × 10-11 kg |
How to write standard index form?
Check whether the given number is less than 1 or between 1 and 10 or greater than 10.
1. If number less than 1
If number is less than 1, then move the decimal point to the right until there is only one digit on left
side of the decimal point.
Count the number of decimal places shifted to right
Then write its product with 10-n
Here, n is the number of places the decimal point has shifted to right.
Example 1: Write 0.327 in standard index form.
0.327 is less than 1
So shift the decimal point to the right side to make the number upto ones place only.
i.e. if decimal is shifted one place to the right then it becomes 3.27
Count the number of digits the decimal point moved to right which is 1
So write the product of 3.27 and 10 with a negative power of 1 or 10-1
∴ Standard index form of 0.327 is 3.27 × 10-1
Example 2: Write 0.0005467 in standard index form.
0.0005467 is less than 1
Shift decimal point to right by 4 places so that 5 comes at ones place i.e. 5.467
So write the product of 5.467 and 10 with a negative power of 4 or 10-4
∴ Standard index form of 0.0005467 is 5.467 × 10-4
2. If number lies from 1 to 10
If number is equal to 1 or between 1 and 10, then product of the given number and 100 is written to make its standard form.
Write 4 in standard index form.
4 lies between 1 and 10
So write the product of 4 and 10 with power of 0 or 100
∴ Standard index form of 4 is 4 × 100
3. If number greater than 10
When the numbers are very large and are greater than 10, then move decimal point to the left side
upto the digit so that there left only one digit on the left of decimal point.
Write the product of the above number and 10n, where n is counted as the number of places
the decimal point has moved to left.
Example 1: Write 17.52 in standard index form.
17.52 is greater than 1
So shift decimal point to left by 1 place to make the number upto ones place i.e. 1.752
Write the product of 1.752 and 10 with a positive power of 1 or 101
∴ Standard index form of 17.52 is 1.752 × 101
Example 2: Write 321500 in standard form.
321500 is greater than 1
So shift decimal point to left by 5 places to make the number upto ones place i.e. 3.21500
Write the product of 3.21500 and 10 with a positive power of 5 or 105
∴ Standard index form of 321.500 is 3.21500 × 105
or 3.21500 × 105 can be written as 3.215 × 105
