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Question
Simplify the following:
`root(3)(x^4y^2) ÷ root(6)(x^5y^-5)`
Solution
`root(3)(x^4y^2) ÷ root(6)(x^5y^-5)`
= `(x^4y^2)^(1/3) ÷ (x^5y^-5)^(1/6)`
= `(x^(4xx1/3)y^(2xx1/3)) ÷ (x^(5xx1/6)y^(-5xx1/6))` .....(Using (am)n = amn)
= `(x^(4/3)y^(2/3)) ÷ (x^(5/6)y^(-5/6))`
= `(x^(4/3)y^(2/3))/(x^(5/6)y^(-5/6))`
= `x^(4/3 - 5/6)y^(2/3 - (-5/6)` .....(Using (am)n = amn)
= `x^(1/2)y^(3/2)`
= `x^(1/2)(y^3)^(1/2)` .....(Using (am)n = amn)
= `sqrt(x) sqrt(y^3)`
= `sqrt(xy^3)`.
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