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## Exponents

** 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

3^{3}

Hence, 3^{3} 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

= 7^{3}

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

= 2^{2} X 5^{3}

Hence, exponential form of 500 = 2^{2} X 5^{3}.

** 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

= 2^{6} X 5^{3}

Hence, exponential form of 1600 = 2^{6} X 5^{3}

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. 10^{8} because:

100000000 = 10 X 10 X 10 X 10 X 10 X 10 X 10 X 10 = 10^{8}

This short notation 10^{8} means that 10 is multiplied with itself for eight times. 10 is known as the base and 8 is known as the **Exponent**.

10^{8} is called as **Exponential Form** of 100000000

Now this short notation 10^{8} is read as 10 raised to the power of 8,

Or we can also say that;

Eighth power of 10 is 10^{8}

Similarly 256 = 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 = 2^{8}

So here 2^{8} is the exponential form of 256, with base as 2 and exponent as 8.

Lets study some more examples:

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

3

Hence, 3

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

= 7

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.

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

= 2

Hence, exponential form of 500 = 2

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

= 2

Hence, exponential form of 1600 = 2