All you need to know about the 1st Law of Indices|exponents.

If you can't solve this at a glance just by looking at it, then this lesson is for you. I'm going to introduce indices, which are exponents, and we are going to start from the basics, from scratch. Okay So first of all, let's discuss what indices are all about—don't solve any question on that indices or exponents. There are so many rules and regulations that govern it. We are going to start from the first law, and the first law of indices is called the multiplication law. the multiplication law Okay Now, what does this multiplication law state? This multiplication law states that a to the power of M a to the power of n multiplied by a to the power of m is equal to a to the power of m plus n This simply means that we have two numbers that are multiplying each other and both of them have the same base. Alright, all you need to do is choose one of the bases and then add the exponents. Let's go through a practical example. for example, you have 2 to the power of 4 multiplied by 2 to the power of 3 now They have the same base and are multiplying terms; there must be multiplication. Okay, they are multiplying. So all you need to do according to this particular law is pick one of the bases and then add the exponents. So your solution will be 2 to the power of 7 if you are asked to leave your answer in index form Excuse me. Now, let's prove this. Since we don't know that this law exists, if you want to solve this, what we'll write is 2 times 2 times 2 times 2 right into 4 places Multiplied by 2 into 3 places: 2 times 2 times 2, okay Excuse me So to leave it in index form, we are going to choose one to write this sentence two times: 1 2 3 4 5 6 7 2 times 7, which is exactly what we have here. So that will show you that this law is the correct one. Now we are going to solve some questions under this law at the end. No, this is very easy. As I said, if you know this law very well, you should have solved it at a glance. Okay, I decided to bring the questions here, and the last one I'm going to solve is this one. Okay, so I advise you to stick to the end, why we did this Okay, so the first one says we should multiply 2 by 3 squared. Alright, here's how to solve it for question one. These are my solutions. So this is 2 4 3 Multiplied by 3 squared. Okay. This is on the index of Exponents: what you need to do is to first of all change them to the same base. You know what I said a base is, right? So let's put them in the same base. This is already in base 3, which is a sign that it can also be in base 3, right? Now this can be written in base 3 as 3 to the power of 5. Okay, so we can rewrite this as 3 to the power of 5. Multiplied by 3 to the power of 2, recall that all these are asked to leave your answer in exponent form or index form. Okay So this simply means because they are in the same base. You are going to pick one, then we are going to add the powers, that is 5 plus 2. So in index form, the solution is 3 to the power of words. 7 You can see how easy it is, right? So that's it for the first one. Now let's solve the second example Now the second example is what that one says, simplified. 5x squared y Multiplied by 2y squared. Okay. Now we are going to collect like terms in this case. Which are the like terms? 5 and 2 are the numerals, so 5 multiplied by 2. Multiplied by, we have X only 1 X. So we are going to bring in only that. Then multiply by we have Y and Y, so we are going to bring in Y multiplied by Y squared. Now anytime you see Y X without any exponent So it means the exponent or the power is incorrect, right? So why is emptiness Y to the power of 1? So we now see that this is 5 times 2, which gives us 10, then we have X squared. So I'm going to apply the multiplication rule here, this multiplication rule. We have Y and Y with the same base, and they are being multiplied. So we are going to pick one and we are going to add the power of 1 plus 2. So finally, the solution to this will be 10x squared y to the power of 3. Very easy, right? If this is the first time you're coming across this channel and you've watched this far That simply means that you love what I'm doing. Please click the subscription button and the notification bell so I can be notified whenever I upload a new video. Thank you so much. All right. Now, let's solve the third one. The third one says on number three simplify this by Sorry, this multiplication multiplied by this. Okay, so this multiplied by this die. to see for D squared is squared multiplied by minus 5 C squared D cubed e Alright. Now we are going to collect like terms as usual. So it's going to be 2 multiplied by minus 5. Okay multiplied by let's pick C anything concerning C C to the power of 4 multiplied by C to the power of 2 Now multiplied by D squared multiplied by D cubed, right? D squared multiplied by D cube equals D to the fifth power. Then multiplied by e to the power of 2 multiplied by e times e to the power of 4 multiplied by e to the power of 1 Let me phrase that so I don't make a mistake. All right. So this is equal to 2 times minus 5, which gives us minus 10. So we pick one C that is in the same base and they are multiplying. So I'm going to pick one C, and we are going to add the powers. So it's going to be 4 plus 2, right? Then we pick one D as well, and we are going to add the powers 2 plus 3. We are going to pick one e as well, and we are going to add the powers 2 plus 1. So finally the solution is minus 10 C, we are going to add 4 plus 2, which gives us 6. Then D 2 plus 3 gives us 5, and E 2 plus 1 gives us 3, and this becomes our solution. Interesting right now at Awenga. Let's solve that last question. That is this one. So the patient says we should simplify. 5 to the power of 3 multiplied by 6 to 5 a D squared times e to the fourth power. All right First of all, we need to put them in the same base; remember what I said. Make sure you use the same base for your numbers. Okay, so we have 5 times a to the power of 3. Okay, multiplied by 6 to 5 is sent in as 5 raised to the power of 4 Okay, so we have D squared and e to the fourth power. Now it's time to combine like terms; to combine like terms is going to be this expression, 5 to the power of 1, right? So you're going to write 5 to the power of 4 multiplied by 5 to the power of 4. Okay, then multiplied by let's pick a to the power of 3 multiplied by a, which is sent in as a to the power Then we have D squared and e to the fourth power So this is equal to you're going to pick one base, which is 5 We are going to add the powers, which is 1 plus 4, then we have a game. We are going to pick one base and we are going to add the power which is 3 plus 1. Then D does not have a like term, so it remains D squared and does not have a like term. So it's still e to the power of 4. So this simply means 5 to the power of 5 times a to the power of 4. D squared and e to the fourth power, and this becomes our final solution