Short Notes: Exponents and Powers

When the numbers given are very large like 54,32,00,00,000 then it is not easy to read them so we write them in the form of exponents.
Exponents make these numbers easy to read, write, understand and compare.

Exponents

To write the large numbers in short form, we use exponents.
Here 8 is the base, 3 is the exponent and 8³ is the exponential form of 512.
This can be read as "8 raised to the power of 3".

The Expanded form of Natural Numbers

When we write the expanded form of a natural number then it can be written in exponential form.
Example: 247983 = 2 × 100000 + 4 × 10000 + 7 × 1000 + 9 × 100 + 8 × 10 + 3 × 1
= 2 × 10⁵ + 4 × 10⁴ + 7 × 10³ + 9 × 10² + 8 × 10¹ + 3 × 1

Some Important Points to Remember

• (-1)^{odd number} = (-1)
• (-1)^{even number} = (1)
• a³b² ≠ a²b³
• a²b³ = b³a²

Laws of Exponents

1. How to multiply powers with the same base?

If we have to multiply the powers which have the same base then we have to add the exponents.
a^m × aⁿ = a^{m + n}
Example: 8³ × 8⁴ = 8^{3 + 4} = 8⁷

2. How to divide powers with the same base?

If we have to divide the powers which have the same base then we have to subtract the exponents.

Example:

3. How to take the power of a power?

If we have to take the power of a power then we have to multiply the exponents.
(a^m)ⁿ = a^mn
Example: (8³)⁴ = 8^{3 × 4} = 8¹²

4. How to multiply the powers with the same exponents?

If we have to multiply the powers where the base is different but exponents are same then we will multiply the base.
a^mb^m = (ab)^m
Example: 8³4³ = (8× 4)³ = 32³

5. How to divide the powers with the same exponents?

If we have to divide the powers where the base is different but exponents are same then we will divide the base.

Example

6. Numbers with Exponent Zero

Any number with zero exponents is equal to one irrespective of the base.
a°= 1
Example: 8° = 1

7. Numbers with Exponent One

Any number with one as the exponent is equal to the number itself.
a¹ = a

Example: 8¹ = 8

8. Power with a Negative Exponent

Negative exponents can be converted into positive exponents.

Example:

Miscellaneous Examples

Example 1:
 
Example 2:

Expressing Large Numbers in the Standard Form

If we have to write very large numbers then to make them easy to read and understand we can write them in the standard form using decimals and exponents from 1.0 to 10.0.
85 = 8.5 × 10 = 8.5 × 10¹
850 = 8.5 × 100 = 8.5 × 10²
8500 = 8.5 × 1000 = 8.5 × 10³
8500 = 8.5 × 10000 = 8.5 × 10⁴ and so on.