What is the simplest form ? `root(4)(x^5)/root(8)(x^4)`  

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lemjay | High School Teacher | (Level 3) Senior Educator

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`root(4)(x^5)/root(8)(x^4)`

`= (xroot (4)(x))/root(8)(x^4)`

Rationalize the denominator. So, multiply the top and bottom by `root(8)(x^4)` .

`=(xroot(4)(x))/root(8)(x^4) * root(8)(x^4)/root(8)(x^4)`

`= (xroot(4)(x)*root(8)(x^4))/ root(8)(x^4*x^4)`

`= (xroot(4)(x)*root(8)(x^4))/ root(8)(x^8)`

`= (xroot(4)(x)*root(8)(x^4))/ x`

Cancel common factor.

`=root(4)(x)*root(8)(x^4)`

Since the radicals have different indices, let's determine the
LCM of 4 and 8, which is 8.

So multiply index of `root (4)(x^5)` by 2 and raise its radicand to a power of 2.

`= root(8)((x)^2)*root(8)(x^4)`

`= root(8)(x^2)*root(8)(x^4)`

Now that the radicals have the same index, proceed to multiply.

`=root(8)(x^2*x^4)`

`= root(8)(x^6)`

Express the index and the exponent of radicand as fraction and reduce it to its lowest term. The lowest term of `6/8` is `3/4` .

`= root(4)(x^3)`

Hence, `root(4)(x^5)/root(8)(x^4)=root(4)(x^3)` . 

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