How do you rationalize the denominator and simplify root3((3a^8)/(4b))?

2 Answers
Tazwar Sikder
May 19, 2017

frac(root(3)(6 a^(8) b^(2)))(2 b)

Explanation:

We have: root(3)(frac(3 a^(8))(4 b))

= frac(root(3)(3 a^(8)))(root(3)(4 b))

= frac(root(3)(3 a^(8)))(root(3)(4 b)) cdot frac(root(3)(4 b))(root(3)(4 b)) cdot frac(root(3)(4 b))(root(3)(4 b))

= frac(root(3)(48 a^(8) b^(2)))(4 b)

= frac(root(3)(48) cdot root(3)(a^(8)) cdot root(3)(b^(2)))(4 b)

= frac(root(3)(2 cdot 2 cdot 2 cdot 6) cdot root(3)(a^(8)) cdot root(3)(b^(2)))(4 b)

= frac(root(3)(2 cdot 2 cdot 2) cdot root(3)(6) cdot root(3)(a^(8)) cdot root(3)(b^(2)))(4 b)

= frac(2 root(3)(6 a^(8) b^(2)))(4 b)

= frac(root(3)(6 a^(8) b^(2)))(2 b)

Barney V.
May 19, 2017

color(blue)((root3(6a^8b^2))/(2b)

Explanation:

root3((3a^8)/(4b)

:.=((3a^8)/(4b))^(1/3)

:.=(3^(1/3)a^(8/3))/(4^(1/3)b^(1/3))

:.=(root3(3)root3(a^8))/(root3(4)root3(b))

:.=root3(3a^8)/(root3(4b))

:.=root3(3a^8)/(root3(4b)) xx (root3(4b))/(root3(4b))xx (root3(4b))/(root3(4b))

:.=root3(3a^8*4b*4b)/(4b)

:.=root3(4b) xx root3(4b) xxroot3(4b) =4b

:.=(root3(2*2*2*2*3a^8b^2))/(4b)

:.=root3(2) xx root3(2) xx root3(2)=2

:.=(cancel2^1root3(6a^8b^2))/(cancel4^2b)

:.=color(blue)(root3(6a^8b^2)/(2b)

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