x is an integer greater than 1. Write the following in order of size, starting with the largest. $x^0,\; x^{-\frac{1}{2}},\; x^{\frac{3}{4}},\; x^{\frac{3}{2}}$ Answer ...

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Worked solution

1. Compare the indices and arrange the powers from largest to smallest.

Since $x$ is an integer greater than 1, $x>1$ . For the same base greater than 1, the larger the index, the larger the value.

Compare the indices:
$4>\frac32=\frac32>\frac34>0=0>-\frac12>-\frac32$

Therefore,
$x^4>x^{\frac32}=x^{\frac32}>x^{\frac34}>x^0=x^0>x^{-\frac12}>x^{-\frac32}.$

Answer

$x^4,\, x^{\frac32},\, x^{\frac32},\, x^{\frac34},\, x^0,\, x^0,\, x^{-\frac12},\, x^{-\frac32}$

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