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:

...

## See

This Answer NowStart your **subscription** to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

Already a member? Log in here.

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**.