1)square root(6x^2y^4) times (3x^5y)^1/2

If that is

sqrt[6(x^2)(y^4)]*[3(x^5)y]^1/2

and you know that sqrt(a) is also a^(1/2), then,

= [(x)(y^2)sqrt(6)]*[(x^2)sqrt(3xy)]

= (x^3)(y^2)sqrt(6*3xy)

= (x^3)(y^2)sqrt(9*2xy)

= 3(x^3)(y^2)sqrt(2xy) ------------answer.

or, simplifying further,

= 3sqrt(2) *(x^3.5)(y^2.5)

= 3sqrt(2) *x^(7/2) *y^(5/2) ---------answer also.

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2.(cube root(42c^4d^17)) / (6cd^11)^1/3

You know that cubrt(a) = a^(1/3)...

= cubrt[(42 c^4 d^17) /(6 c d^11)]

= cubrt[7 c^3 d^6]

= cubrt(7) *(c)(d^2) -------------------------answer.

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3.((6x^5y^7)^1/2 times (square root(3x^2y^4)) / square root(2x)

= sqrt(6 x^5 y^7) *sqrt(3 x^2 y^4) / sqrt(2x)

= sqrt[(6 x^5 y^7)(3 x^2 y^4) / 2x]

= sqrt[18 x^7 y^11 / 2x]

= sqrt[9 x^6 y^11]

= 3(x^3)(y^5)sqrt(y) --------answer

or

= 3(x^3)(y^(11/2)) ------answer also.

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We used:

---(a^b)^c = a^(b*c)

---a^b *c^b = (a*c)^b

---a^b / a^c = a^(b-c)

---a^b * a^c = a^(b+c)