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If x^{1/5 }> x^{1/3 }, then how many of the following statements are definitely true about x?

x^{2 }> x^{3}

x^{1/3 }> x^{4}

1 > x^{1/3 }> x^{-3}

x^{-1/3 }> x^{3}

- 1
- 2
- 3
- 4

2 votes

Best answer

x^{1/5 }> x^{1/3}

Dividing by x^{1/3 }on both sides we get

x^{1/5 - 1/3} > 1

x^{-2/15} > 1

x^{2/15} < 1

So, x is less than 1. Here, x can be either < -1 or between 0 and 1.

- x
^{2 }> x^{3}

Is always true - x
^{1/3 }> x^{4}

Is true only for positive x. If x is < -1, this becomes false as (-1)^{4}turns out to be even. - 1 > x
^{1/3 }> x^{-3}

Is not true for positive x as x^{-3}> x^{1/3} - x
^{-1/3 }> x^{3}

Is always true

So, 1, and 4 are true. 2, 3 are false.

So, answer is 2. 2