 Hello and welcome to this session. In this session we will discuss cubes. Let's see what are perfect cubes or cube numbers. The numbers obtained when a number is multiplied by itself three times are known as cube numbers or you can also say perfect cubes. Now we have that 8 can be written as 2 into 2 into 2. We have multiplied 2 to itself three times so as to get 8. So we say this 8 is a perfect cube. Then again the numbers 729,000, 1,728 are also perfect cubes. Then we have cubes of even numbers are even and cubes of odd numbers are odd. But if you consider the even number 4, now its cube is 64 which is also even. Thus we say that cube of even numbers are even. Now let's consider odd number 5. Its cube is 125 which is odd. Thus we have cubes of odd numbers are odd. Consider this table. If the units digit of a number is 1 then the units digit of the cube of the number would also be 1. If the units digit of a number is 2 then the units digit of the cube of the number would be 8. Then units digit of a number is 3. Then the units digit of the cube of the number would be 7. Units digit of a number is 4. Then the units digit of the cube of the number would also be 4. If the units digit of a number is 5 then the units digit of the cube of the number would be 5. Then if the units digit of a number is 6 then the unit digit of the cube of the number would be 6. Then we have unit digit of a number is 7 then unit digit of the cube of the number would be 3. If the unit digit of a number is 8 then the unit digit of the cube of the number would be 2. If the unit digit of a number is 9 then the unit digit of its cube would be 9. Then if the unit of a number is 0 and the unit of the cube of that number would also be 0. Now we discussed some interesting patterns. First we have adding consecutive odd numbers. Observe this pattern of sums of odd numbers. Now consider the first line of this pattern which is 1 equal to 1 equal to 1 cube. Now as you can see to obtain the sum as 1 cube we need one odd number and that is 1. Now observing the second line of this pattern you see that to obtain the sum as 2 cube we need 2 consecutive odd numbers and that is 3 and 5 which is equal to 8 that is 2 cube. Then according to the next line of this pattern we have that to obtain 3 cube as the sum we have added 3 consecutive odd numbers 7, 9 and 11 so as to get 27 which is 3 cube and this pattern goes on in the same way. So if we are asked that how many consecutive odd numbers we would need to obtain the sum as 10 cube then our answer would be 10 consecutive odd numbers. So if you are asked the number of consecutive odd numbers needed to obtain the sum as 10 cube then we would simply say we would need 10 consecutive odd numbers so as to get the sum as 10 cube according to this pattern. Next we discuss cubes and their prime factors. Consider the number 4 its prime factorization is given as 2 into 2. Now consider cube of 4 which is equal to 64 its prime factorization is given as 2 into 2 into 2 into 2 into 2 into 2. That is we have multiplied the prime factor 2 by itself 6 times. We could also write this as 2 cube into 2 cube that is each prime factors appears 3 times in its prime factorization. Thus we say that if in the prime factorization of any number each factor appears 3 times then the number is a perfect cube. Let's consider the number 400 its prime factorization is given as 2 into 2 into 2 into 5 into 5. Now let's make triplets of these prime factors. So this is one triplet. Now as you can see the number 2 is left alone and this 5 into 5 is one pair it is not a triplet. As you can see each factor does not appear 3 times in the prime factorization of the given number 400. So we say that 400 is not a perfect cube. Next we discuss smallest number of prime factorization of the multiple that is the perfect cube. Let's consider the number 72. Prime factorization of 72 is equal to 2 into 2 into 2 into 3 into 3. Now let's make the triplets of these prime factors. So as you can see we just get one triplet and this is one pair 3 into 3. This is not a triplet. Thus the 72 is not a perfect cube. Now for 72 to be a perfect cube each factor should appear 3 times but as you can see the factor 3 is appearing just 2 times. So to make 72 perfect square we need to multiply the 72 by 3 so that we have that each factor appears 3 times. Thus we say 72 into 3 is equal to 216. Now this 216 is a perfect cube. Thus we say smallest natural number by which 72 should be multiplied to make a perfect cube is 3. Next we discuss cube roots. We know that finding the square root of a number is the inverse operation of squaring. So we say that finding cube root is the inverse operation of finding cube. Like we have 3 cube is equal to 27. So we say cube root of 27 is 3. Now this symbol is used to denote the cube root. So we say cube root of 27 is 3. Now we shall discuss cube root through prime factorization method. Consider the number 216. Now prime factorization of 216 is given as 2 into 2 into 2 into 3 into 3 into 3. Now let's make triplets of these prime factors. So this is one triplet and this is the other triplet. Now cube root of 216 is given by taking out one factor out of each triplet. So from this triplet we take out a 2 multiplied by 3 from this triplet. So we get that cube root of 216 is equal to 2 into 3 that is 6. So this is how we can find the cube root of any number through prime factorization method. Next we have cube root of a cube number. If we know that a given number is a cube number then we have the following method which we will discuss now. Let's consider the number 175616. Now this is a cube number. Let's try and find out the cube root of this number. Now in the first step we consider the number and then we make groups of 3 digits starting from the right most digit of this number. So we get this is one group and this is the other group and as you can see each group is of 3 digits. Now in the next step we consider this first group that is 616. Now this group would give us the ones digit or you can say units digit of the required cube root. As you can see that the units digit of this group is 6 and we know that if a number ends in 6 then the cube root of that number would have 6 as the units digit. So we say that once digit or you can say the units digit of the required cube root that is the cube root of the given number is 6. Now let's consider the other group that is 175. We already know that 5 cube is 125 and 6 cube is 216. Also we know that 175 is less than 216 and it is greater than 125. Now to get the 10th digit of the required cube root we consider the smaller number out of these two numbers and we know that 125 is smaller than 216 and as you can see that the ones digit of 125 is 5. We take this as the 10th digit of the required cube root. So we have 10th digit of the required cube root is 5. Now we have got the ones digit of the required cube root which is 6 and the 10th digit of the required cube root which is 5 and so we say that cube root of the number 175616 is equal to 56. This is how we find the cube root of a perfect cube or a cube number. So this completes the session. Hope you have understood the concept of cubes and cube roots.