According to one of the laws of indices, if we have exponents with similar bases dividing each other, we have to subtract the index/exponents
$$i.e,\ \frac{x^p}{x^q}=x^{p-q}$$
Statement 1: p==q+5
p-q=5
$$\frac{x^p}{x^q}=x^{p-q}=x^5$$
This does not give us a definite answer because we don't know the value of x. Hence, statement 1 is INSUFFICIENT.

Statement 2: $$x^q=32$$
This does not provide information about the value of p. So, we cannot use this to find the value of $$\frac{x^p}{x^q}$$.
Hence, Statement 2 is INSUFFICIENT.

Combining statement 1 and 2 together,
$$p=q+5$$
$$x^q=32$$
Note: x and q are positive integers. So, we can have 2 positive scenarios; x=2; q=5 and x=32; q=1
1st scenario; if x=2 and q=5
Then, p=q+5 =5+5=10
$$x^{p-q}=2^{10-5}=2^5=32$$

2nd scenario; if x=32 and q=1;
p=q+5 = 6
$$x^{p-q}=32^{6-1}=32^5=33554432$$
Since we cannot ascertain the definite answer, the combined statement are INSUFFICIENT.

Answer = OPTION E

BTGmoderatorDC wrote:$$If\ x,\ p\ and\ q\ are\ positive\ integers,\ then\ \frac{x^p}{x^q}\ =\ ?$$

1. p = q + 5

$$2.\ x^q\ =\ 32$$

\[x,p,q\,\,\, \geqslant \,\,1\,\,\,{\text{ints}}\]
\[? = {x^{\,p - q}}\]
\[\left( {1 + 2} \right)\,\,\,\,\,\left\{ \begin{gathered}
\,Take\,\,\left( {x,q,p} \right) = \left( {32,1,6} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = {32^5}\,\,\,\, \hfill \\
\,Take\,\,\left( {x,q,p} \right) = \left( {2,5,10} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = {2^5} \ne {32^5} \hfill \\
\end{gathered} \right.\]

The above follows the notations and rationale taught in the GMATH method.