Prove that: `root4(x times x^2timesx^3 times x^4)` `<=(x +x^2+x^3+x^4)/4`   (4th)sqrt(x1*x2*x3*x4)<=(x1+x2+x3+x4)/4 please help. the first term is a fourth square root

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to write the left side using the property of exponentials such that:

`root(4)(x*x^2*x^3*x^4) = root(4)(x^(1+2+3+4))`

`root(4)(x*x^2*x^3*x^4) = root(4)(x^10)`

Converting the radical into a power yields:

`root(4)(x^10) = x^(10/4) => root(4)(x^10) = x^(5/2)`

`root(4)(x^10) = sqrt(x^5) = sqrt(x^4*x) = x^2sqrtx`

Hence, you need to prove the following inequality such that:

`x^2sqrtx =< (x + x^2 + x^3 + x^4)/4`

`4x^2sqrtx =< x + x^2 + x^3 + x^4`

You need to raise to square both sides such that:

`16x^5 =< (x + x^2 + x^3 + x^4)^2`

`16x^5 =< x^2 + x^4 + x^6 + x^8 + 2(x^3 + x^4 + x^5 + x^5 + x^6 + x^7)`

`16x^5 =< x^2 + x^4 + x^6 + x^8 + 2(x^3 + x^4 + 2x^5 + x^6 + x^7)`

`16x^5 =< x^2 + 2x^3 + 3x^4 + 4x^5 + 3x^6 + 2x^7 + x^8`

Hence, checking if the given inequality holds yields that `16x^5 =< x^2 + 2x^3 + 3x^4 + 4x^5 + 3x^6 + 2x^7 + x^8` , which proves the inequality.

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