First, let's clarify the values involved. The expressions \( \sqrt{-7} \) and \( \sqrt{-8} \) represent complex numbers. Specifically:
\[ \sqrt{-7} = i\sqrt{7} \] \[ \sqrt{-8} = i\sqrt{8} = 2i\sqrt{2} \]
Both are imaginary numbers, and \( \sqrt{-7} \) lies along the imaginary axis at \( i\sqrt{7} \), and \( \sqrt{-8} \) is at \( 2i\sqrt{2} \).
To find a rational number with a denominator of 7 between these two imaginary numbers, we need a rational number of the form \( \frac{a}{7} \) where \( a \) is some integer. However, since \( i\sqrt{7} \) and \( 2i\sqrt{2} \) are imaginary numbers, we cannot find a rational number that lies between them on the real number line.
If you mean a rational number on the real line that expresses something related to these two complex numbers, please clarify the context or the definition of "between" you're using, as rational numbers do not ordinarily compare with imaginary numbers in the same manner.