Radical expressions with higher roots | Algebra I | Khan Academy

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Not long ago – we dealt with roots and we had We only used the square root And we saw that if I wrote the root code like this and put it Underneath it 9, this means I want the square root of 9 It is positive 3 Or I could tell the positive square root of 9 What is implicitly understood from this is that I want Take the square root I can write it like this And I can write the root code like this and Putting the 2 here, which also means the square root Square root of number 9 Find a number if you square it I will get 9 And root code no Applies only in the case of square root The value below the root can be changed and taken Root of any number For example, if I want to ask you, how much – you can Imagine what is called a cube root, or it can be called With the third root of 27 how much? Well, it is any number I raise to the power of 3 I get 27 Well, the only number if you raise it to power 3 I will get 27 is 3 3 x 3 x 3 = 27 That is 9 x 3 = 27 So let me solve another one So if we have 16– I’ll do this in a different color If we have 16 and I want to find its quadruple root So how many times if I multiplied it by myself 4 times would I get 16? And if you don’t realize this quickly, you can 16 to be analyzed into its main factors Until you find it Let’s see that 16 = 2 x 8 8 is 2 x 4 And 4 is 2 x 2 That is, this equals the quadratic root of 2 x 2 X 2 x 2 We have four from number 2 here Well, we have four times the number 2 multiplied together, so the quadratic root It will be 2 This can be viewed as The square root of if it’s all -2 You will also succeed . Just as you multiply the square roots, you should Quadruple roots But the root symbol means the root root With this, we have simplified Traditional square roots And now I hope you can simplify the roots It has higher root strength So let’s try several examples Let’s say I want to simplify this phrase The fifth root of 96 As I said earlier, let’s analyze the value of its factors So 96 is 2 x 48 And 48 = 2 x 24 24 is 2 x 12 12 = 2 x 6 And 6 is 2 x 3 So the fifth root equals 2 x 2 x 2 2 x 2 x 2 x 2 x 2 x 2 x 2 X 3 Or in another way, it can be written as a picture Fractional forces You can consider them as fractional forces And we’ve already talked about this That is, it equals 2 x 2 x 2 x 2 x 2 x 3 ^ 1/5 Let me make this clear If we want to extract the n root of a number, it is equal to taking This number is to the power of 1 / n These two phrases are equal So if we take this number of the force 1/5, then this is Same as 2 x 2 x 2 x 2 X 2 ^ 1/5 X 3 ^ 1/5 I now have numbers multiplied by each other I have 2 times multiplied by itself 5 times And I raised it to the force 1/5 Well, 1/5 is 2 Or, the fifth root will equal 2 So we have 2 here This will be 3 ^ 1/5 (2 x 3) ^ 1/5 You can also simplify it But if we want to keep it as a root, we can write it Thus: 2 x the fifth root of 3 Let’s try another issue We will try another issue I will use variables now Let’s say we want to simplify the sixth root of 64 X x ^ 8 Let’s start with 64 first 64 = 2 x 32, and 32 is 2 x 16 16 is 2 x 8 And 8 is 2 x 4 And 4 = 2 x 2 So we have 1, 2, 3, 4, 5, 6 We have 2 ^ 6 That is, the value equals the sixth root of 2 ^ 6 This is the 64– x x ^ 8 Now, the sixth root of 2 ^ 6 direct This part is equal to 2 It will be 2 x sixth root For x ^ 8 x ^ 8 How can this be simplified? Well, x ^ 8 is equal to x ^ 6 X x ^ 2 We have the same basis, so we collect the foundations Which is equal to x ^ 8 This equals 2 x the sixth root of x ^ 6 X x ^ 2 And the sixth root, this part, the sixth root For x ^ 6, is x So this equals 2 x x x The sixth root of x ^ 2 Now, we can simplify this further if I thought about it And remember, this phrase Tied (x ^ 2) ^ 1/6 And if you remember the properties of the foundations, it’s when Increases the number of a given power, and then raises the whole amount Another power, it’s equal x ^ 2 x 1/6 Or– let me write this– 2 x 1/6, Equal – I shouldn’t forget that I have 2x here So I have 2x here and 2x here It is the same as 2x X x ^ 2/6 Or, if we want to write it simpler or lower Commonly, we’ll get 2x x x ^ – what do we have Here? x ^ 1/3 If you want to write it as a root, you can write it So: 2 x 2x x the third root of x Or think about it another way, you could say– So we can go from this point We can write this We can ignore this, as we did before And we could say, that this equals 2 x (x ^ 8) ^ 1/6 This equals 2 x x ^ – 8 x 1 / 6– x ^ 8/6 Now we can do a shortcut to fraction So the amount is 2 x x ^ 4/3 This and this are equal Why? Because we have 2 x x or 2 x x ^ 1 X x ^ 1/3 We add 1 to 1/3, so we get 4/3 I hope you found this lesson Have fun I think it is useful and you will find it as the main factor So if I wanted to find the sixth root, it would be Ali Find the basic factor that is repeated six times Then I can find the value of 2 ^ 6 In any case, I hope you found this explanation useful

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