solve cubed root x equals 8

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That is:

^3\sqrt x = 8

or:

x^{(\frac13)} = 8

Cube each side to give:

(x^{(\frac13)})^3 = 8^3

Use the product rule, then the 3 will cancel out.

(x^{(\frac{1}{\cancel{3}})})^{\cancel{3}} = 8^3

x^1 = 8^3