 Hello and welcome to the session. In this session we will discuss a question which says that simplify the first four terms in the binomial expansion of 1-3x whole raised to power 3 by 5 calculate the value of x for which 1-3x whole raised to power 3 by 5 becomes equal to 0.64 raised to power 3 by 5 and hence find the value of 0.64 raised to power 3 by 5 correct to five significant figures. Now before starting the solution of this question we should know our result and that is the expansion of a binomial expression which is 1 plus x whole raised to power n is equal to 1 plus nx plus n into n minus 1 over 1 into 2 into x square plus n into n minus 1 into n minus 2 over 1 into 2 into 3 into x cube plus so on up to infinity where the absolute value of x is less than 1 and n be any fraction or a negative integer. Now this will work as the key idea for solving out this question. Now we will start with the solution. Here we have to expand 1 minus 3x whole raised to power 3 by 5 for expanding this we will use the result given in the key idea. Here x is minus 3x and n is 3 by 5 so expanding this by the binomial expansion 1 minus 3x whole raised to power 3 by 5 will be equal to 1 plus nx and here is 3 by 5 and x here is minus 3x plus n into n minus 1 that is 3 by 5 into 3 by 5 minus 1 over 1 into 2 into x square x here is minus 3x square plus into n minus 1 into n minus 2 which will be 3 by 5 into 3 by 5 minus 1 into 3 by 5 minus 2 over 1 into 2 into 3 into x cube which is minus 3x whole cube plus so on which is equal to 1 minus 9 by 5x plus 3 by 5 into 3 by 5 minus 1 is minus 2 by 5 over 1 into 2 is 2 into minus 3x whole square is 9x square plus 3 by 5 into 3 by 5 minus 1 is minus 2 by 5 into 3 by 5 minus 2 is minus 7 by 5 upon 1 into 2 into 3 is 6 into minus 3x whole cube will be minus 27x cube plus so on further this will be equal to 1 minus 9 by 5x and here 2 will be cancelled with 2 and on calculating this this will be equal to minus 27 by 25x square and here 2 into 3 is 6 and 3 will be cancelled with 3 so it will be minus I am calculating this 189 over 125x cube plus so on therefore we have got 1 minus 3x whole is 2 by 3 by 5 is equal to 1 minus 9 by 5x minus 27 by 25x square minus 189 over 125x cube plus so on so this is the expansion of 1 minus 3x whole is 2 by 3 by 5 up to 4 terms and let us name it as 1 now we have to find the value of x for which 1 minus 3x whole is 2 by 3 by 5 becomes equal to 0.64 raised to power 3 by 5 so now we have to find the value of x for which 1 minus 3x whole raised to power 3 by 5 becomes equal to 0.64 raised to power 3 by 5 this means 1 minus 3x whole raised to power 3 by 5 is equal to 0.64 whole raised to power 3 by 5 this implies 1 minus 3x is equal to 0.64 which implies 1 minus 0.64 is equal to 3x this implies 0.36 is equal to 3x and which implies 0.12 is equal to x now substituting this value of x in 1 so now we will put the value of x as 0.12 in 1 now here 1 minus 3x whole raised to power 3 by 5 on calculating and putting the value of x as 0.12 will become 0.64 raised to power 3 by 5 therefore the left hand side of 1 will become 0.64 raised to power 3 by 5 is equal to the right hand side will be 1 minus 9 by 5 into 0.12 minus 27 by 25 into 0.12 square minus 189 by 125 into 0.12 whole cube which is equal to 1 minus 9 by 5 into 0.12 minus 27 by 25 into 0.0144 minus 189 by 125 into 0.001728 now this will be equal to on calculating 1 minus 1.08 by 5 minus 0.3888 by 25 minus 0.326592 over 125 on solving further this will be equal to 1 minus 0.216 minus 0.015552 minus 0.002612736 which will be equal to 1 minus 0.234164736 further this is equal to 0.765835264 which is equal to 0.76584 correct to 5 significant figures hence 0.64 raised to power 3 by 5 is equal to 0.76584 correct to 5 significant figures so this is the solution of this question that's all for this session hope you all have enjoyed this session