Trying something new

• Radicals


  The number inside the radical symbol
• Index


  The number that indicates the root, e.g. 2 for square root, 3 for cube root
• You can only add or subtract radicals if they have the same index and the same radical
• Square root examples


  • 3√2 + 5√2
  • 5√3 - 2√3
• Cube root examples


  • 8√27 - 3√27
  • 4√64 + 2√64
• Simplifying square roots


  1. Find two numbers that multiply to give the radicand, where one is a perfect square

  2. Put the perfect square outside the radical and multiply by the other number
• If radicals do not have the same index and radicand, you cannot add or subtract them, you can only write them together
• The final answer for example 6 is 4√6 - 4√5
• t outside you multiply it by -2 so 2 * -2 oh sorry -2 * 2 so once again the square of 4 that is 2 you put it outside then you multiply it by -2 so that would be -2 * 2 then copy squ of 6 plus the square of 9 that is three you put it outside radical symbol so copy three and then copy s of 5 so once again three came from the square of 9 square of 9 that is three you put it outside class the radical symbol then simply copy square of 5 minus the squ of 9 that is three you put it outside the radical symbol then simply copy square of 6 all right and then you simplify this one 3 * 2 that is is 6 then capy of 5 -2 * 2 that's -4 then simply copy TK of 6 once again negative * positive is negative 2 * 2 is 4 copy of 6 Plus simply copy this one 3 5 - 3 6 all right and then you arrange class can we combine this one this 6 5 - 4 6 nope we cannot combine this one because they don't have the same radicans this is five and this is six so therefore we can only combine this one and this one class 44 of 6 and -3 6 all right so you need to arrange class okay you need to arrange this one combine like terms so this will be 6 5 + 3 5 - 4 of 6 - 3 of 6 all right and this will be B so you combine this one 6 + 3 that is 9 then simply copy squ of 5 and for this one -4 - 3 so negative combine with negative copy the sign then add so once again if you have4 -3 both negative simply copy negative sign then you add the numbers so that would be -7 all right so -4 - 3 that's -7 copy < TK of 6 that's it class that's the answer there for number six answer for number six class all right so the answer for number six that would be 9 of 5 - 7 6 so we don't have enough space class just pause the video this will be the answer for number six class we don't have enough space so let's try number seven so I will be erasing this one cl since we don't have enough space we'll be answering number seven all right so number seven we have 3 > 8 x Cub + 4x < TK 18x - 3x < 2x all right so this will be so simplify this one so copy three then for eight we can simplify this one by 4 * 2 right four is a perfect square then 2 is not so that would be 4 * 2 then for X Cub we can Factor this out by X2 * X right why is that sir because we can divide 2 by two here so once again class for this x Cub we Factor this out by x^2 * X because we can divide this two by the index and then copy 4X then for 18 we can Factor this out by 9 * 2 right 9 * 2 that is 18 9 is a perfect square then 2 is not for X so we cannot factor x so simply copy X understood that there's one here we cannot divide 1 by two so simply copy X then copy - 3x of 2x so we cannot simplify 2x class so simply copy so this will be three then the square Ro of 4 that is two you put it outside radical symbol of 4 that is 2 you multiply it by 3 so 3 * 2 then the square of X2 that is X so the square of x square that is X because we can divide 2 by two here for the index so 2 / by 2 that is 1 so you put it outside radical symbol the square root of x s that is X then 2 will remain inside radical symbol okay once again two will remain inside the radical symbol because we don't have the square root of two as well as X so the square root of x we don't have the square root of x we cannot divide this one because understood that there's one here here 1 / 2 we cannot divide this one so X will remain inside the radical symbol so copy square root of 2 will remain as well as X so copy + 4 x then theun of 9 that is three you put it outside radical symbol so multiply it by 4X so 4X * 3 then theun of 2 we don't have so two will remain as well as X so capy s of 2x then minus 3x of 2x so this will be multiply 3 * 2 that is 6 copy x 3 * 2 that is 6 * X that is 6 x copy s of 2x + 4 * X that is 4X * 3 that is 12x copy of 2x - 3x of 2x all right and this will be so you add this one 6 x 12 x and - 3x combine CL 6 x + 12 x - 3x so 6X + 12 x - 3x then simply copy squ of 2x so once again class adding algebraic expression if you still remember just simply add the numbers so 6 + 12 that is 18 - 3 that is 15 then copy X class copy the literal coefficient so simply add the numbers then copy the literal coefficient copy X class so that would be 15x so 6 + 12 that is 18 - 3 that is 15 then simply copy X so 15 x then copy s Ro of 2x that's it Le that's the answer for number seven all right so let's try number eight for the cube root I hope you're still with me class for this one addition and subtraction of radicals so number eight so we have 2 cubot of 2 + 3 cubot of 2 by the way class could you try this one number eight this I I think this will be very easy for your class number eight could you try this one and you put your answer in the comment section down below let me check if you really understand our topic this number eight class is very easy for you I think so so you try this one class and you put your answer in the comment section down below let me check class if you really understand our topic class just try this one class number eight so let's try number nine class let's just try number nine class because number nine and 10 are more complicated than number eight so let's have 4 cube root of 16 I'm sorry 5 cube root of 16 + 2 cubot of 54 - 8 cubot of two all right so they don't have the same radicans therefore we need to simplify class now this remember class the cube root the perfect cubes so let's start with eight right eight is a perfect Cube so we have8 because 8 we can simplify this one by 2 ra to power of three right 8 is a perfect cube root because 2 ra to power of 3 once again CL 2 ra to power of 3 it means 2 * 2 * 2 and 2 * 2 * 2 that is 8 so therefore 8 is a perfect Cube because 2 * 2 * 2 that is8 what's the cube root of 8 class that's correct the cube root of 8 that is two and we have 27 that would be 3 ra to the power of 3 the cube root of 27 that is 3 because 3 * 3 * 3 that is 27 and let's have 64 that would be 4 raised to the power 3 because 4 * 4 * 4 that is 64 let's just stop on 8 27 and 64 CL all right so we have five cube root of so we don't have cube root of 16 you think of a number class that when you multiply it by itself three times you will get 16 nope there is none so we need to simplify this one 16 you find two numbers that one number is a perfect Cube the other is not that when you multiply you will get 16 so we can simplify the 16 by 8 * 2 right because 8 is a perfect Cube then two is not okay check L 8 is a perfect cube the cube root of 8 that is 2 and 8 * 2 that is 16 all right so once again you find two numbers class one number is a perfect Cube the other is not and 8 * 2 that is 16 so 8 is a perfect Cube two is not then copy plus sign then two cube root of 454 class okay once again you find two numbers that one number is a perfect Cube the other is not so for 54 we can simplify this one by 27 * 2 right 27 is a perfect Cube then 2 is not when we multiply we will get 54 so once again the cube root of 27 that is three all right right so then copy Min - 8 cubot of 2 once again class I hope you're not confused with a cube root class to simplify you find two numbers one number is a perfect Cube the other is not that when you multiply you will get this number all right and this will be five then the cube root of 8 that is two you put it outside radical symbol once again plus the cubot of 8 that is 2 because 2 * 2 * 2 that is 8 so you put it outside rical symbol you multiply it by five so that would be 5 * 2 then copy cube root of 2 then plus copy two the cube root of 27 that is three you put it outside the radical symbol once again the cubot of 27 that is 3 because 3 * 3 * 3 that is 27 then you multiply it by two cube root of 27 is three you put it outside you multiply it by two then copy cube root of 2 2 then - 8 cubot of 2 all right so you multiply this one 5 * 2 that is 10 cubot of 2 + 2 * 3 that is 6 cubot of 2 then - 8 cubot of 2 all right and combine like terms so combine this one so 10 + 6 that is 16 - 8 that is 8 so 10 + 6 that is 16 - 8 that is 8 then copy cube root of two that's it that's the answer there for number 9 8 cube root of two all right so last one class last one example number 10 so I hope you're still with me our video is quite long but we need to answer this one class number 10 last one example class do not forget class to answer number eight and you put your answer in the comment section down below all right let's have this one number 10 3x cube root of 24x + 5 Cub root of 81 x ra^ 4 so capy 3x then cube root of 24 class can we simplify 24 yep that would be 8 * 3 right 8 * 3 three 8 is a perfect Cube then three is not when we multiply we will get 24 once again class you find two numbers that one number is a perfect Cube the other is not and8 is a perfect Cube three is not when we multiply we will get 24 for X class we cannot simplify X because this is one we cannot divide 1 by 3 so simply copy X Plus then copy five cube root of so for 81 class can we simplify 81 yep we can simplify 81 by 27 * 3 right let's check this 27 * 3 3 * 7 is 21 1 carry 2 3 * 2 is 6 + 2 is 8 81 so 27 * 3 and 27 is a perfect Cube then 3 is not when we multiply we will get 81 and for X ra the^ of 4 we can simplify this one by X ra^ of 3 * X X because if we combine This One X Cub * X that would be X ra to^ 4 all right once again class if we combine this one if you multiply same base simply copy X then you will add exponent 3 + one that is four now question sir why do we need to have X Cub sir we need to have X Cub so that we can divide 3 by 3 okay we can divide 3 by 3 so that we can put it outside the radical symbol so there will be x ra^ 3 then x rais the^ 1 all right so once again class I hope you remember this one class in simplifying radicals so this will be 3 x copy 3x the cube root of 8 that is two you put it outside the radical symbol so 3x * 2 then copy the cube root of so we cannot simplify three as well as X so therefore 3x will remain inside the radical symbol so that would be 3x cubot of 3x + 5 the cube root of 27 that is 3 you put it outside radical symbol so 5 * 3 then for three cube root of 3 we don't have cube root of 3 so three will remain inside rical symbol for X Cub we can divide 3 by 3 3 / by 3 that is 1 so we can cancel this out and then for X you put it outside radical symbol so time x then for this one we cannot divide 1 by 3 so X rais the^ of 1 will remain inside the radical symbol so copy cube root of 3x and this will be simplify 3x * 2x that is 6X cube root of 3x then for this one 5 * 3 that is 15 * X that is 15 x + 15x Cub root of 3x so can we add now the radical yep we can add the radicals because they have the same radicans and they have the same index so combine this one 6 x + 15x so once again 6X + 15x simply copy cube root of 3x and this will be 6 x + 15 x so once again add the number first 6 + 15 that is 21 then simply copy x copy cube root of 3x that's it plus that's the answer there for number 10 so I apologize class if our video is quite long and I hope you learned something new today once again if you learn something new today do not forget to like share and subscribe you share to your friends class and to your classmates so that we can help more students once again this is teacher MJ have a great day class goodbye for now bye-bye