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Laws of Exponents Negative Exponents

Sometime we are encountered with large numbers like 100000000, so for keeping such numbers simple and short we convert them to short notations i.e. 108 because:
100000000 = 10 X 10 X 10 X 10 X 10 X 10 X 10 X 10 = 108
This short notation 108 means that 10 is multiplied with itself for eight times. 10 is known as the base and 8 is known as the Exponent.

108 is called as Exponential Form of 100000000

Now this short notation 108 is read as 10 raised to the power of 8,
Or we can also say that;
Eighth power of 10 is 108

Similarly 256 = 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 = 28
So here 28 is the exponential form of 256, with base as 2 and exponent as 8.

Lets study some more examples:




Example 1: Convert 27 into exponential form:
Solution: Given number is 27
And with prime factorization we get:
27 = 3 X 3 X 3
Since 3 is repeated three times, so we get
33
Hence, 33 is the exponential form of 27.




Example 2: Express 343 as a power of 7
Solution: Given number is 343
And with prime factorization we get:
343 = 7 X 7 X 7
Since 7 is repeated three times, so we get
= 73
Hence, 343 is the third power of 7.




Till now you have understood that a large number can be expressed in the form of short notions which contains one base and one exponent.
But sometimes there also comes a situation where a large numbers is expressed in the form of short notions which contains more than one base and more than one exponent.

Example 3 : Express 500 into exponential form:
Solution: Given number is 500
And with prime factorization we get:
500 = 2 X 2 X 5 X 5 X 5
Since 2 is repeated twice and 5 is repeated thrice, so we get
= 22 X 53
Hence, exponential form of 500 = 22 X 53.




Example 4 : Express: 16000 into exponential form:
Solution: Given number is 16000
And with prime factorization we get:
16000 = 2 X 2 X 2 X 2 X 2 X 2 X 2 X 5 X 5 X 5
Since 2 is repeated six times and 5 is repeated thrice, so we get
= 26 X 53
Hence, exponential form of 1600 = 26 X 53


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