# Simplify: `(27x^3 y^6z^9)^(1/3)`

justaguide | College Teacher | (Level 2) Distinguished Educator

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The expression `(27x^3 y^6 z^9)^(1/3)` has to be simplified

`(27x^3 y^6 z^9)^(1/3)`

=> `3*x*y^2*z^3`

The simplified form of `(27x^3 y^6 z^9)^(1/3) = 3*x*y^2*z^3`

donatzm | College Teacher | eNotes Newbie

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Taking the cube root of a product, means that we can take the cube root of each of the pieces and multiply them together again.  That is, the cube root of 27*x^3*y^6*z^9 is the same as the cuberoot(27)*cuberoot(x^3)*cuberoot(y^6)*cuberoot(z^9)

27 = 3*3*3=3^3, so cuberoot(27)=3

Another way to write cuberoot(27) is 27^(1/3), so we can see this another way: cuberoot(27)=cuberoot(3^3)=(3^3)^(1/3)=3^(3*1/3)=3^1=3

So we use the same notation for finding the cuberoots of x^3, y^6, and z^9:

cuberoot(x^3)=(x^3)^(1/3)=x^(3*1/3)=x^1=x

cuberoot(y^6)=(y^6)^(1/3)=y^(6*1/3)=y^2

cuberoot(z^9)=(z^9)^(1/3)=z^(9*1/3)=z^3

Multiplying all these together we get:

cuberoot(27*x^3*y^6*z^9)=3*x*y^2*z^3