Hello everyone in this video we are going to learn how to solve radicals or how to simplify roots. This is a very important topic for class 9th and class 10th students and this is a single video in which you are going to learn how to add, subtract, multiply and divide radicals. We will also be solving few examples at the end of the video. So let's get started.
First we'll learn the addition of roots or addition of radicals. If you have two radicals as root 5 plus root 5 5 then the addition of these two radicals will give you 2 root 5. This you can consider same as the addition of variables. If x plus x is there then you add and write 2x.
Similarly if root 5 and root 5 is there then you can add it directly and write 2 root 5. If the numbers in the radical signs are same then you can add them directly. The next one is 2 root 3 plus root 3. So this This is same as 2x plus x will give you 3x. 2 root 3 plus root 3 will give you 3 root 3. The next one is 3 root 5 plus 4 root 3. Now here we have different numbers under the radical.
So you cannot add them. So you have to write it as is like 3 root 5 plus 4 root 3. It will be the same. You cannot add the numbers if you have different numbers under the root sign.
So this is all about the addition of roots. Next we will move on to subtraction of roots. Subtraction and addition follows the same root.
If the numbers under the roots are same we can directly subtract and add the numbers. If the numbers are different we cannot add and subtract. This is root 5 minus root 5. This you can consider same as x minus x.
Root 5 minus root 5 will be 0. Next one is 3 root 2 minus root 2. So this is 3x minus x or 3 root 2 minus 1 root 2. 3 minus 1 is 2. and root 2 will be there 3 root 2 minus root 2 will be 2 root 2. The next example let us change it so the next example is 4 root 5 minus 3 root 3 here we see that the numbers under the root sign are different we cannot subtract the numbers we have to write it the same as is so 4 root 5 minus 3 root 3. Let's move on to the multiplication of roots before we go ahead I would like to introduce you the that root 2 is equal to 2 to the power half. Root 2 and 2 to the power half is same. Root 2 is equal to 2 to the power half when we write it in exponent.
So the power of 2 is half. So by using this exponent thing we can write root 2 as 2 to the power half times 2 to the power half. Now here we are going to use the rules of exponent when the bases are same and multiplied the powers get added.
This will be 2 to the power half. half plus half which is equal to 2 to the power 1 equal to 2. So root 2 times root 2 is equal to 2. If the numbers under the root sign are same and we multiply same numbers under the root sign we will get the number itself. Next we will move on to the next example which is root 5 times 2 root 5. This you can consider same as x times 2x.
2 times root 5. 5 times root 5 so root 5 and root 5 will get multiplied and that will be equal to 2 times 5 which will give you 10 so 2 root 5 times 5 is equal to 10 because there are 2 root 5 and root 5 times root 5 will give you 5 that is getting multiplied by 2 so this is equal to 10. Next we have 3 root 2 and 4 root 3. multiplication the numbers outside the root get multiplied first so this will be 3 times 4 and the numbers inside the root get multiplied so this will be 2 times 3 so the answer will be 3 times 4 is 12 and 2 times 3 is 6 12 root 6 so the numbers under the root gets multiplied separately and the numbers outside the root get multiplied let's now learn division of roots When you divide any number with the same number you will get 1. We can understand it with the help of exponent so root 3 you can write as 3 to the power half divided by 3 to the power half. And by the rule of exponent when two numbers get divided with the same base the exponents are subtracted. So this will be 3 to the power half minus half which is equal to 3 to the power 0 which is equal to 1. anything to the power 0 is equal to 1 root 3 divided by root 3 is equal to 1 next one is 2 root 5 divided by root 5 so here root 5 and root 5 gets cancelled or divided and we are left with only number 2 so the answer is 2. The next one is 3 root 2 divided by 2 root 3. So here we see that we cannot cancel out the numbers outside or the numbers inside.
So this will be the same as 3 root 2 divided by 2 root 3. One more thing I want to tell you that we cannot leave root in the denominator for that there is a process to change the denominator into a number which is without roots that process is called rationalization of roots that we are going to learn in our further videos. For now we have completed the division of two roots. Let us now solve few questions to understand these rules clearly so here we are going to solve few examples the first one is 3 times root 2 plus root 3 we multiply both the numbers with the outside number first we will multiply 3 by root 2 and then we multiply 3 by root 3 and the sign will be what is there in between the two numbers so this will be 3 root 2 then there is plus sign and then 3 root 3 this is our final answer for this one sorry this is 3 as the as the numbers inside the roots are different we cannot solve it further in the next one too we are going to multiply root 5 by first number and then by the second number 2 root 5 times root 5 then there is plus sign in between plus root 3 times root 5 so 2 root 5 times root 5 will be 2 times 5 because root 5 times root 5 is 5 plus root 3 times root 5 will be root 3 times 5 which is equal to 2 times 5 is 10 plus root 15. This is our final answer we are going to box the answer. The next one is 3 root 2 times 3 minus root 2 in parenthesis. So here too first we will multiply it with the first number and then the second number and the sign will be the same as the sign in between the two numbers.
So this will be 3 root 2 times 3 minus 3 root 2 times root 2. The outside number gets multiplied with the 3. So 3 times 3 is 9 root 2 will be there minus 3 and this is root 2 times root 2 which will give us 2. So this will be 9 root 2 minus 6. This is our final answer. Next one is 3 plus root 2 in parenthesis multiplied multiplied by 2 minus root 2 in parenthesis so here too we are going to use the same rule first we will multiply 3 with both the numbers in the second parenthesis and then we will multiply 2 with both the numbers in the second parenthesis so this will be 3 times 2 minus sine is minus in between the two numbers 3 times root 2 Then there is plus. root 2 times 2 so this will be 2 root 2 it is a general practice to write root 2 after the whole number so i have written 2 root 2 next one is negative sign is there in between 2 root 2 times root 2 so this will be root 2 times root 2 now 3 times 2 is 6 minus 3 root 2 plus 2 root 2 minus root 2 times root 2 is to Now 6 minus 2 is 4 minus 3 root 2 plus 2 root 2. This is same as minus 3x plus 2x is minus x minus root 2. So this is our final answer. We are going to box the answer. This was the whole video about simplification of roots.
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