 Hello and welcome to the session. In this session we will discuss binomial theorem for negative integer or fractional index, approximations and evaluation of root. First of all let us discuss binomial theorem for negative integer or fractional index. Now the binomial theorem holds hold even when the index is a negative integer or a fraction. So for all values of n negative or fractional or both 1 plus x whole raised to power n is equal to 1 plus nx plus n into n minus 1 the whole over 1 into 2 into x square plus n into n minus 1 the whole into n minus 2 the whole over 1 into 2 into 3 into x cube plus so on up to infinity where mod of x is less than 1. Now there are some points to remember when n is a negative integer or a fraction. Now we have discussed this as the binomial expansion of 1 plus x whole raised to power n where n is a negative integer or a fraction and where x is a real number. Now we know that the binomial theorem when n is a positive integer and x and y are real numbers is given by this result which is x plus y whole raised to power n is equal to summation and cr into x raised to power n minus r into y raised to power r where r varies from 0 to n but when n is a negative integer or a fraction and in that case the symbols nc0, nc1, nc2 and so on are not used as they become meaningless because they have been in only when n is a positive integer and secondly this expansion contains infinite number of terms and the expansion is valid only when x is numerically less than 1 that is the second term in the binomial expression 1 plus x all left inside is less than 1 numerically that is when mod of x is less than 1 that is the second term on the binomial expression on the left inside is less than 1 numerically and this condition is absolutely essential. Now let us discuss this with the help of an example. Now let us expand 1 minus x whole raised to power minus 3 when x is equal to 3 that is we have taken x numerically greater than 1 so by using the binomial expression when n is a negative integer 1 minus x whole raised to power minus 3 will be equal to 1 plus nx that is 1 plus minus 3 into minus x the whole plus n into n minus 1 that is minus 3 into minus 3 minus 1 the whole whole upon 1 into 2 into x square that is minus x whole square plus so on. Now we have taken x is equal to 3 therefore for x is equal to 3 it will be 1 minus 3 whole raised to power minus 3 which is equal to 1 plus minus 3 the whole into minus 3 the whole plus minus 3 into minus 4 the whole into minus 3 whole square whole upon 2 plus so on further this implies minus 2 raised to power minus 3 is equal to 1 plus 9 plus 6 into 9 plus so on which is an absurd as minus 2 whole raised to power minus 3 cannot be equal to this expression so from this we can conclude that whenever x is numerically equal to or greater than 1 the expression may be invalid that is in this case we have taken x is equal to 3 which is numerically greater than 1 and we have applied the binomial expression when n is a negative integer and we have got a result which is as absurd therefore we can say that for the value of x which is numerically equal or greater than 1 the expansion is invalid so the expansion is valid only when more of x is less than 1 that is whenever x is numerically less than 1. Now in the expansion of a plus x whole raised to power n the first term should be made unity if it is different from 1 that is a plus x whole raised to power n will be equal to a raised to power n that is taking a raised to power m common it will be a raised to power n into 1 plus x over a whole raised to power n and then the expansion of this is valid only when the numerical value of x over a is less than 1 that is more of x over a is less than 1 a plus x whole raised to power n can be written as now taking x raised to power n common it will be x raised to power n into 1 plus a over x whole raised to power n and the expansion of this is valid only and now let us discuss approximations now in the expansion of 1 plus x whole raised to power n which is equal to 1 plus nx plus n into n minus 1 the whole over 1 into 2 into x square plus n into n minus 1 the whole into n minus 2 the whole over 1 into 2 into 3 into x cube plus so on what of x is less than 1 now as x is less than 1 so the terms of this expansion go on decreasing and if x is very small a stage may be reached when we may neglect the terms containing higher powers of x of the expansion thus if x be so small that it squares and higher powers may be neglected then whole raised to power is equal to 1 plus nx so if x is so small that it squares and higher powers may be neglected then 1 plus x whole raised to power n is equal to 1 plus nx that is the terms containing the squares and the higher powers may be neglected when x is so small so 1 plus x whole raised to power n is equal to 1 plus nx so this value is approximate value 1 plus x whole raised to power n now let us discuss an example and that is if x be so small that it squares and higher powers may be neglected find the approximate value of 1 minus 3 by 2x whole raised to power minus 3 now the approximate value 3 by 2 into x whole raised to power minus 3 is now using this result of approximation so the approximate value of 1 minus 3 by 2x whole raised to power minus 3 is 1 plus n which is minus 3 into minus 3 by 2 into x which is equal to 1 plus 9 by 2 into x that is when the x is so small that it squares and higher powers may be neglected then the approximate value of 1 minus 3 by 2 into x whole raised to power minus 3 is 1 plus 9 by 2 into x and now let us discuss evaluation of a root now suppose we have to find the pth root of any number q then we express the number q in the form x raised to power p plus y where x raised to power p is nearest to the number q at y which is either positive or negative is very small in comparison x raised to power p therefore q raised to power 1 by p is equal to x raised to power p plus y whole raised to power 1 by p which can be written as x raised to power p whole raised to power 1 by p into 1 plus y over x raised to power p whole raised to power 1 by p now on expanding 1 plus y over x raised to power p whole raised to power 1 by p by using the binomial expansion and approximate value of the root. That is the approximate value of this can be obtained. Now let us discuss an example. In this we have to find the square root of 901 up to 5 decimal places. Now for finding the square root of 901 first of all we will write 901 as 900 plus 1. Therefore 901 can be seen as now 901 can be seen as 30 raised to power 2 plus 1. So we have written 901 in the form x raised to power 3 plus y that is 30 square plus 1 where 30 square which is equal to 900 is nearest to 901 and 1 is very small in comparison to 30 raised to power 2. Now the square root of 901 can be seen as 901 whole raised to power 1 by 2 which is further equal to 30 raised to power 2 plus 1 whole raised to power 1 by 2. Now 30 raised to power 2 plus 1 whole raised to power 1 by 2 will be equal to now taking 30 raised to power 2 common so it will be 30 square whole raised to power 1 by 2 into 1 plus 1 over 30 raised to power 2 whole raised to power 1 by 2 which is further equal to 30 into. Now for this we will apply the binomial expansion now where n is 1 by 2 and x is 1 by 30 square. So it will be 1 plus nx that is 1 plus 1 by 2 into 1 by 30 square plus n into n minus 1 the whole whole upon 1 into 2 into x square that is 1 by 30 square whole square plus so on. Now this is further equal to 30 into 1 plus 1 over 2 into 9 into 10 square plus 1 by 2 into minus 1 by 2 into 1 by 2 into 1 by 9 into 9 into 1 by 10 raised to power 4 plus so on. Further this is equal to 30 into 1 plus 1 over 18 into 10 raised to power 2 minus 1 over 8 into 1 over 81 into 1 over 10 raised to power 4 plus so on which is equal to 30 into 1 plus now on solving this we will get 0.00055 minus and on solving this we will get 0.000154 plus so on. Further on solving this this is equal to 30 into 1.00549846 the whole which is further equal to 30.0164953 h which is equal to 30.01650 correct to 5 decimal places therefore the square root of 901 is equal to 30.01650 correct 5 decimal places. So for the evaluation of root we will proceed with the steps which are mentioned in this example. So in this session you have learnt about binomial theorem for negative or fractional index and then the approximations and then evaluation of a root. So this completes our session hope you all have enjoyed the session.