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.

answer: the correct answer is -32

step-by-step explanation:

answer would be 8.64 or s = 11π/4 inches

step-by-step explanation:

s = r θ  

where s = arc length  

r = radius = 3 inches  

θ = central angle in radians = ( π /180)*165  

s = 3(π/180)*165  

= 8.64 inches