If x^7 < x^6 , which of the following could be the value of x ?

Number properties often can be verified with testing.

And if questions have numbers with large exponents, usually the question can be replicated on a smaller scale.

We can test values using smaller exponents as long as we stay true to the original problem (inequality).

The prompt gives an inequality that means "a number raised to an odd power is smaller than the number raised to an even power."
Thus: \(x^3 < x^2\)

Which of the following COULD be the value of x?

I. −3
Test: \((-3)^3=-27\) and \((-3)^2=9\)
\(-27<9\) ... Possible

Number property tested is that
-- A negative number raised to an odd power is negative
-- Raised to an even power, the number is positive

II. 2/3
Test the much easier number \(\frac{1}{2}\) INSTEAD (any fraction between 0 and 1 works)
\((\frac{1}{2})^3=\frac{1}{8}\), and
\((\frac{1}{2})^2=\frac{1}{4}\)

\(\frac{1}{8}<\frac{1}{4}\). ...Possible

Property: For any fraction between 0 and 1, the value of the fraction decreases as the exponent increases.

III. 3/2 - Test:
\((\frac{3}{2})^3=\frac{27}{8}=3.xx\), and
\((\frac{3}{2})^2=\frac{9}{4}=2.xx\)

LHS is not < RHS. Not possible

Property: For any number greater than 1, the value of the number increases as the exponent increases

I and II only

Answer C