Inverted 7th chords in open position

Question

Im working through a music theory exercise book and I just need confirmation if I am not understanding something or if the book made a mistake again(there are quite a few mistakes in this workbook which isn't too good for me). I recreated the example they've given:

Just a quick reminder that this dominant 7th chord is in open position.Anyways, the first measure is supposedly the root position(it has V^7 below it in the book), the 2nd measure is the first inversion (V^65), 3rd measure is second inversion(V^43) and the last measure is 3rd inversion (V^42)

I understand that to figure out an inversion, I must look at the lowest note and basically compare it to the notes in root position. From bottom up in the root position, it is established that the chord is G, B(natural), F and D. However if I compare it to the chord in the third measure --which happens to be D, B(natural), G, F--I get that this chord position is third inversion since D(the 7th) is the lowest note. So am I understanding a concept incorrectly or is did the book make a mistake? If I am correct, then that means the third and fourth measure are switched.

Oh and I forgot to mention that this chord is in C minor though I don't think it really relates to this question too much.

Answer 1

I think it will help you more to look at the chords in close position to get an understanding of how the chord itself, G7, is built and what the inversions are and why.

The picture above is equivalent in nature to the one in your example, but for simplicity the chords are in close position.

Now let's look at the leftmost chord which is G7 in root position. We see the notes G, B, D, and F and notice they are all built in thirds. The root is the base of the chord and can be looked as our reference point when giving the other notes in the chord intervals. Because we build chords in thirds and the root is G, then we know that B is a third, D is a fifth, and F is a seventh based on how intervals are named. We call this root position because the root is the lowest note in the chord.

The second chord has the exact same notes, but we took the G that was in the bass and we moved it an octave up. Now the B, the third of the chord, is in the bass. We call this first inversion because it's the first note we encounter after the root in the chord.

The third chord has the exact same notes, but we took the B that was in the bass and we moved it an octave up. Now the D, the fifth of the chord, is in the bass. We call this second inversion because it's the second note we encounter after the root in the chord.

The last chord has the exact same notes, but we took the D that was in the bass and we moved it an octave up. Now the F, the seventh of the chord, is in the bass. We call this third inversion because it's the third note we encounter after the root in the chord.

Answer 2

The book didn't make a mistake.

Think of the root inversion as the zeroth inversion - or as I prefer not an inversion at all. Then the chord in measure 2 is the first inversion, measure 3 is the 2nd inversion and measure 4 is the 3rd inversion.