Hi everyone welcome to smart math online tutor through this video! We are going to study about indices. First, let us see what is indices! Look at this value?

This is a valid written in index notation!

This is read as 2 to the power 3 and it can be written as 2 into 2 into 2 2 to the power 3 is called the index form, whereas 2 into 2 into 2 is called the expanded form? So the value of 2 to the power 3 can be obtained by multiplying 2 into 2 into 2, which will give you the answer as 8. Now in an index form, there are few parts you should know now here. This is 2 to the power 5 5 is called the index, whereas 2 is called the base. The whole thing together is called a power! Now, let’s see how to write a number as a product of powers of prime numbers! Look at this example write 300 as a product of powers of prime numbers? For this. What we do is we divide 300 by the prime numbers! Until you obtain one, let’s get started 300 the smallest prime number is 2! As the last digit in 300 is 0!

We know 300 can be divided by 2 without a remainder. So we do that? Then the answer becomes 150 again it can be divided by 2 and then it is 75 75. You know it can be divided by 3. So let’s divide it by 3 to get the answer as 25 and 25 can be divided by 5, so we divided by 5 and you get the answer as 5 and once again we divided by 5 in order to obtain 1?

As the final answer right now, 300 can be written as 2 into 2 into 3 into 5 into 5? These numbers i have taken from here right now: 2 is written twice. 5 is written twice. So therefore we can write 2 and 5 in index notation like this. So here we have written 300 as a product of powers of prime numbers. Now, let’s move on to another section, we are going to see how to express a power of a product as a product of powers!

Now you have to clearly understand these two terms: power of a product and a product of powers? Look at this 3 into 2 whole thing to the power 2.

This is a power of a product. A product 3 into 2 is a product! It is raised to a power, so we call this as a power of a product. In order to expand this, we write the same bracket twice, because the power is two. Now we can remove the bracket and write twos together and threes together like this! Now since 2 is there twice and 3? is there twice? we can write it in index notation like this now here 2 to the power 2 into 3 to the power 2, two powers are multiplied, so this part we call as a product of powers and in very simple notation. We can write it like this 3 into 2 whole thing to the power 2. Without the brackets, we have written it as 2 to the power 2 and 3 to the power 2. . So it’s very simple, no need to write the expanded form?

Always that is this part is not necessary. These two lines is for explanation: it’s not very necessary. You can directly write the answer like this right. Moving on to the next part, expressing a product of powers as a power of a product. Now the reverse of the previous case. Now look at this 8 a to the power 3 8 a to the power 3! You know, 8 is 2 to the power 3, so you can write 8 as 2 to the power 3 into a to the power 3. . Now both have the power or the index as 3! Therefore you can write this like this! With the help of a bracket, i have taken the index outside of a bracket, so then 2 into a you know it is a product and a product is raised to the power of 3.

So this has been now written as a power of a product in simplified form. You can write it as 2a whole thing to the power 3. . Remember here, you must use brackets.

Otherwise the answer will be wrong, so you can write 8 a to the power 3 as 2a whole thing to the power 3.

. So this is how you write a product of power as a power of a product right then we are going to another important section. That is the power of a negative integer.

Now, first look at this one here it is 2 to the power 3 2 to the power.

3 2 is the integer here, but this is not negative. First, let’s see how to write power of positive integers and then, let’s move on to negative integers, so 2 to the power 3 can be expanded as 2 into 2 into 2 and your answer becomes 8. This is very simple and you can understand it very simply and let’s look into another example: 5 to the power 2. . It is 5 into 5 and the answer is 25. Now you see when the integer is positive, the answer is always positive, or the volume of this power is always positive. So we can conclude the value of a power of a positive integer is always positive. You have to remember this now. Let us see what happens when the integer is negative. Look at this minus 2 to the power 3.

So once when you write the expanded form, it becomes minus 2 into minus 2 into minus 2?

? Now minus 2 into minus 2, the first pair minus 2 into minus 2! You know it is plus 4, because minus into minus is plus and plus 4 into minus 2 becomes minus 8. Then our answer is minus 8 right now. Do you see when the integer is negative? the answer has become negative and there is another speciality. Remember the inte. When the integer is negative, the answer will become negative. Only if the index over here or the power is the odd number right! So that’s a very important thing. The value of an odd power of a negative integer is always negative!

If the power is odd, the answer will be negative! You know odd numbers, 1, 3, 5, 7 and so on!

So if the power is odd, answer is negative. Let’s see what, if the power is even now look at this minus 5 to the power 2. , so in that case power is 2. , so when you expand it becomes minus 5 into minus 5.

, you know, 5 into 5 is 25 minus into minus is plus, so the answer is plus 25? Now, in this case, since the power is even the answer is positive, so you have to remember this very clearly.

If the power is odd, the answer is negative. If the power is even answer is positive right. So this is a very important section. Now look at this question fill in the blank with less than greater than or equal sign! So let us see the blank minus a to the power 5 blank minus a to the power 4, so we are going to fill in this blank in order to fill in this blank. Since we don’t know the values of a a is an unknown term, so we look into the powers look at 5! It is an odd power. Therefore, the value of minus a to the power 5 will be a negative value, whereas 4 is an even power, so the value of that part will be a positive value. Now you know in a number line? Negative values are smaller than positive values!

Therefore, with that knowledge, we can complete this blank like this, so that tells minus a to the power. 5 is less than minus a to the power 4! Fine, then this is the end of the session.

Hope i made myself clear with indices and powers, so see you with another smartmath clip until then. Goodbye. .