Exponent or Power:
It is a number which indicates the number of times multiplication is to be performed. It is written above the base number on its right side.An exponent can be a positive or negative number.
Example: 2³ means we need to multiply 2 for three times i.e. 2³ = 2 x 2 x 2. Here, 2 is known as the base number and 3 is known as the exponent.

Powers with Negative Exponents:
For any non-zero integer a, a^-m = 1/ a^m, where m is a positive integer. Also, a^-m is known as the multiplicative inverse of a^m.
Example: 2^-2 = 1/(2 x 2) = 1/2².
Here, 2^-2 is the multiplicative inverse of 2².

Exponential Laws:
For any non-zero integers a & b, and for any integer’s m & n, we have the following laws of exponents:

(1) a^m / aⁿ = a^m-n
Example: Solve 2⁴ / 2³.
Solution: Using the law of exponent, we can write 2⁴ / 2³ = 2^4-3 = 2¹ = 2.

(2) (a^m)ⁿ = a^mn.
Example: Solve (2²)³.
Solution: Using the law of exponent, we can write (2²)³ = 2^2×3 = 2⁶ = 64.

(3) a^m x b^m = (ab)^m
Example: Solve 2² x 3².
Solution: Using the law of exponent, we can write 2² x 3²= (2 x 3)² = 6² = 36.

(4) a^m / b^m = (a/b)^m
Example: Solve 6² / 3².
Solution: Using the law of exponent, we can write 6² / 3²= (6/3)² = 2² =4.

(5) a⁰ = 1
Example: Solve 6⁰.
Solution: Using the law of exponent, we can write 6⁰ = 1.

Using exponents to express small or large numbers:
There are many practical situations where we need to write down the number which may be very large or small. At such place, we can use exponents to express them in small numbers.

1. Standard form to write natural numbers of the form abc000000……
Step 1: For any given number, count the number of digits appearing after the left-most digit.
Step 2: For expressing it in standard form, write down the left-most digit first.
Step 3: If the original number have digits other than zero after the left-most digit then put a decimal point. Write down all those digits until zero comes. And if there are no digits after left-most digit, then skip this step.
Step 4: Place a multiplication sign and write down the counted digits in step-1 as exponent to base number 10.

Example: Express 120000000000 in exponent form.
Solution: In standard form, the number 120000000000 will be written as 1.2 x 10¹¹.

2. Standard form to write decimal numbers of the form 0.00000…..abc:
Step 1: For any given number, count the number of digits starting from the decimal point till the right-most last digit.
Step 2: If there is only one digit at the end of the given number, then simply write down that digit. Place a multiplication sign and write down the counted digits in step-1 with a negative sign as exponent to base number 10.
Step 3: If there are two or more non-zero digits at the end of number. Then, write down the right-most digit followed by a decimal point and the other non-zero digits.
Step 4: Calculate the number of digits in step-1, minus the number of digits appearing in step-3 after decimal point.
Step 5: Place a multiplication sign and write down the counted digits in step-4 with a negative sign as exponent to base number 10.

Example: Express 0.000000000000012 in exponent form.
Solution: In standard form, the number will be written as 1.2 x 10^-14.

3. Comparison of large and small numbers using exponent:
Step 1: Write down all the given numbers into the standard form.
Step 2: Convert the numbers so that they have the same exponents.
Step 3: Compare numbers other than exponents and decide the large and small numbers.

Example: Compare the two numbers 1.5 x 10¹⁰ and 397.6 x 10⁸.
Solution: We will make power of both the exponents to 8, so we get the two numbers as 150 x 10⁸ and 397.6 x 10⁸.
On comparing, we can see that 397.6 x 10⁸ is larger than 1.5 x 10¹⁰.