 Hello and welcome to the session. In this session first we will discuss powers with negative exponents. We already know that if you have a number say 2 to the power m this is equal to 2 multiplied by 2 multiplied by 2 multiplied by 2 that is we multiply 2 m times then we have for any non-zero integer a a to the power minus m is equal to 1 upon a to the power m where this m is a positive integer and this a to the power minus m is called the multiplicative inverse of a to the power m. For example, multiplicative inverse of 2 to the power minus 4 is given by 1 upon 2 to the power 4. Next we shall discuss laws of exponents. We take a and b the non-zero integers and we have m and n are any integers then we have falling laws of exponents according to which we have a to the power m multiplied by a to the power m is equal to a to the power m plus n. Then the next law is a to the power m upon a to the power n is equal to a to the power m minus n. Next law is a to the power m whole to the power n is equal to a to the power m n then we have a to the power m multiplied by b to the power m is equal to a b whole to the power m then the next one is a to the power m upon b to the power m is equal to a upon b whole to the power m then we have one more law according to which we have a to the power 0 is equal to 1. We also say that a upon b whole to the power minus m is equal to b upon a whole to the power m. Let's simplify the expression minus 4 to the power 5 divided by minus 4 to the power 8 this could be written as minus 4 to the power 5 upon minus 4 to the power 8. So for this we will use this law of exponent which is a to the power m upon a to the power m is equal to a to the power m minus m. So this becomes equal to minus 4 to the power 5 minus 8 that is minus 4 to the power minus 3 this is equal to 1 upon minus 4 to the power 3 that is we have 1 upon minus 64. So the value for the given expression is equal to minus 1 upon 64. Next we discuss use of exponents to express small numbers in standard form. Very small numbers can be expressed in standard form using negative exponents. Let's consider the small number 0.0000021. Let's express this number in the standard form. This could be further written as 21 upon 1 000 000. That is we put 7 times 0 since there are 7 digits after the decimal. This is further written as 21 upon 10 to the power 7 or we can also write it as 2.1 multiplied by 10 upon 10 to the power 7 that is we get 2.1 multiplied by 10 to the power minus 6. So this is how we have expressed the given number which is very small in the standard form using negative exponents. Consider the number 3.02 multiplied by 10 to the power minus 6. Let's express this number in the usual form. This would be equal to 3.02 upon 1 000 000 that is we put 6 times 0. Now as you can see we have 6 zeros in the denominator. So in the numerator we move this decimal 6 places to the left side. So we get 0.00000302. So this is how we have expressed this number in the usual form. This completes the session. Hope you have understood the concept of exponents and powers.