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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?

Respuesta :

luisejr77

Answer:

For [tex]x^x > 500,000[/tex]  [tex]x=7[/tex]

For [tex]x^{(-x)} > 500,000[/tex]  [tex]x=-7[/tex]

Step-by-step explanation:

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

[tex]x^x > 500,000[/tex]

We tested with x = 6

[tex]6^6=46,656[/tex]

[tex]46,656 > 500,000[/tex]

Inequality is not met because [tex]46,656 < 500,000[/tex]

We test with the following integer x = 7

[tex]7^7=823,543[/tex]

[tex]823,543 > 500,000[/tex]

Then the smallest positive integer that will make [tex]x^x > 500,000[/tex] is 7

Inequality is met.

In the same way the largest negative integer that will make [tex]x^{(-x)} >500000[/tex] is [tex]x=-7[/tex] Beacuse [tex]7^{-(-7)}=823,543[/tex]

Answer:

Smallest positive integer value for [tex]x^x>5000[/tex] is,

x = 7,

Largest negative integer value for [tex] x^{-x} >500000[/tex] is,

x = -8

Step-by-step explanation:

If [tex]x^x>500000[/tex]

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

Case 1 : If x < 7,

[tex]x^x < 500000[/tex]

Case 2 : If x ≥ 7,

[tex]x^x > 500000[/tex]

Hence, smallest positive integer value of x is 7.

Now, if [tex]x^{-x}>500000[/tex]

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

Case 1 : if x is odd negative integer,

[tex]x^{-x} < 50000[/tex]

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,

[tex]x^{-x} < 500000[/tex]

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

(ii) x is less than -8,

[tex]x^{-x} > 50000[/tex]

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

Hence, the largest negative integer value that will make [tex]x^{-x}> 500000[/tex] is x = -8.

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