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) >500000?

Answer 1

Smallest positive integer value for
`x^x>5000` is,

x = 7,

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

x = -8

Explanation:

If
`x^x>500000`

∵ If x is a positive integer then the possible values of x = 1, 2, 3, 4, 5, 6, 7.....

Case 1 : If x < 7,

`x^x < 500000`

Case 2 : If x ≥ 7,

`x^x > 500000`

Hence, smallest positive integer value of x is 7.

Now, if
`x^(-x)>500000`

∵ If x is negative integer then the possible value of x = -1, -2, -3, -4,.....

Case 1 : if x is odd negative integer,

`x^(-x) < 50000`

eg : -1, -3, -5, -7,...

Case 2 : If x is even negative integer then there are further two cases,

(i) x is more than or equal to -6,

`x^(-x) < 500000`

eg x = -6, -4, -2,

(ii) x is less than -8,

`x^(-x) > 50000`

eg : x = -10, -12, -14,...

Hence, the largest negative integer value that will make
`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

`7^7=823,543`

`823,543 > 500,000`

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

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`