 Hello and welcome to this session. In this session we discussed the following question which says evaluate cube root of minus 512 upon 1000. Before moving on to the solution let's recall the cube root of a rational number a upon b. So cube root of a upon b is equal to cube root of a upon cube root of b. This is the key idea to be used in this question. Let's move on to the solution. We need to find the cube root of minus 512 upon 1000. You think the idea we say cube root of minus 512 upon 1000 is equal to cube root of minus 512 upon cube root of 1000. Consider the number 512. Now we will express this number as product of times 2, 256 times is 512 then 2, 128 times is 256, then 2, 64 times is 128, 2, 32 times is 64, 2, 16 times is 32, 2, 8 times is 16, then 2, 4 times is 8 and 2, 2 times is 4, then 2, 1 times is 2. So the number 512 is written as the product of its primes by multiplying 2 to itself 9 times. Now the number minus 512 can be written as minus 2 multiplied by itself 9 times. Now cube root of minus 512 is given by making the groups in triplets of the same prime. So this is one group, this is another group and this is another group. Now we will take out one number from each group. So from this take out minus 2 multiplied by a minus 2 from this group multiplied by a minus 2 from this group and this is equal to minus 8. So we get cube root of the number minus 512 is equal to minus 8. In the same way consider the number 1000, now 2, 500 times is 1000, 2, 250 times is 500, 2, 125 times is 250, 5, 25 times is 125 and 5, 5 times is 25, 5, 1 times is 5. So the number 1000 is written as the product of its primes. So this is equal to 2 into 2 into 5 into 5 into 5. Now again we make the groups in triplets of the same prime. So this is one group, this is another group. Then cube root of 1000 is given by taking out a number from each of these groups and multiplying them together. So we take a 2 from this group, 2 multiplied by a 5 from this group. This is equal to 10. Thus cube root of 1000 is equal to 10. So now cube root of minus 512 upon 1000 is equal to cube root of minus 512 that is minus 8 upon cube root of 1000 that is 10. Now 2, 4 times is 8 and 2, 5 times is 10. So this is equal to minus 4 upon 5. Thus our final answer is cube root of minus 512 upon 1000 is equal to minus 4 upon. Each session hope you have understood the solution for this question.