Prove that if x is an positive real number such that x + x^-1 is an integer, then x^3 + x^-3 is an integer as well.

By closure property of multiplication and addition of integers,

If is an integer

∴ is an integer

From which we have;

is an integer

Step-by-step explanation:

The given expression for the positive integer is x + x⁻¹

The given expression can be written as follows;

By finding the given expression raised to the power 3, sing Wolfram Alpha online, we we have;

By simplification of the cube of the given integer expressions, we have;

Therefore, we have;

By rearranging, we get;

Given that   is an integer, from the closure property, the product of two integers is always an integer, we have;

is an integer and is also an integer

Similarly the sum of two integers is always an integer, we have;

is an integer

is an integer

From which we have;

is an integer.

step-by-step explanation:

just plug in each number for x and see which ones work and don't

hope this !

step-by-step explanation:

d = distance between moon and sun, which in this problem is the hypotenuse of a right triangle.  

the distance y is the side opposite to angle x in the right triangle.   use the definition of the sine function from basic trigonometry.

sin(x) = y/d ⇒

d = y/sin(x)