Question

What is the smallest positive integer that will make x^x > 500,000? What

is the largest negative integer that will make x^(-x) > 500,000?

Answer 1

Smallest positive integer value for
`x^x>500000` is,

x = 7,

Largest negative integer value for
`x^(-x)>500000` is,

x = -8

Explanation:

If
`x^x>500000`

By graphing calculator,

`x>6.83`

Thus, the smallest possible positive integer value of x is 7,

Now,

`x^(-x)>500000`

Possible negative integer values of x are -6, -7 and -8,

If x = -6, -7, and -8,

`(-6)^(6)=46656`

`(-7)^(7)=-823543`

`(-8)^(8)=16777216`

`\because 16777216 > 500000`

Thus, the largest negative integer value of the inequality
`x^(-x)>500000` is,

x = -8.

Answer 2

For
`x^x > 500,000`
`x=7`

For
`x^((-x)) > 500,000`
`x=-7`

Explanation:

We need to find the smallest positive whole number that satisfies the inequality:

`x^x > 500,000`

We tested with x = 6

`6^6=46,656\\\\46,656 > 500,000`

Inequality is not met because
`46,656 < 500,000`

We test with the following integer x = 7

Then we have that:

`7^7=823,543\\\\823,543 > 500,000`

Then the smallest positive integer that will make
`x^x > 500,000` is 7 because Inequality is met.

In the same way the largest negative integer that will make
`x^((-x)) >500000` is
`x=-7` Beacuse
`7^(-(-7))=823,543>500,000`