*In the OAV she does not have these powers without gems.
Of all the characters in the Tenchi Muyo series Ryoko is my favorite. She causes so much trouble for poor Tenchi. Ryoko draws her power from three stones one around her neck and one on each wrist. Ryoko is a demon who until the first episode Ryoko Resurrected was imprisoned in a cave just outside the Masaki Shrine. She was released by Tenchi after his grandfather tricked him into going into the cave. You'd think that she'd be more thankful considering he released her and restored her beauty. But then again this Ryoko after all.
Ryoko is the protagonist in this series. She is an Oni, an ancient Japanese devil. She was created by Washu my using a simple life form called Mass and from one of Washu's egg cells. Because of this Washu considers Ryoko as more of a daughter rather than an invention. When Zero copied Ryoko she absorbed some of her memories. After Zero allowed her emotions to surface, Washu combined Zero and Ryoko. The result was Ryoko with more obvious emotions. Ryoko's chief rival for the affections of Tenchi is Princess Ayeka. Ayeka is the crowned princess of Jurai and has the mannerisms and powers that come with her position. While Ryoko is outgoing and brash, Ayeka is more withdrawn and proper.
Little is known about Ryoko's past before her battle with Yosho. Before and during the battle Ryoko was under the control of Kagato. He took over her after imprisoning Washu and taking Soja. Kagato used her to lead the only successful attack on Jurai. The power in her gems is so similar to that of Jurai's trees that she managed to slip by the Jurain border. During her attack on Jurai she shot down a ship that crashed into a building. Sasami was in that building and fell to her death. Kagato sent Ryoko there to find Tsunami and capture her. She could not find Tsunami and was forced to retreat. Yosho pursued her and eventually caught her on Earth.
After that she spent 700 years in the cave near the Masaki shrine. While trapped in the shine she created a pseudo-gem which she used to fight Tenchi with. That gem was incomplete when Tenchi found her. Tenchi later destroyed it during their fight. While Tenchi was growing up she watched over him and fell in love with him when he got older.
I also have a few screensavers available one of which features Ryoko and is very well done.
Ryoko's gems, which only appeared functionally in the OAV give her some more powers that she does not have access to naturally. Ryoko wears one in each wrist and one in her neck. Recently, I've come across some information that explains the origins of the gems. They contain Washu's goddess power. Over 20,000 years ago, Washu possessed powers equal to that of her two sisters, Tsunami and Tokimi. I'm not sure why this happened but I do know that it was a punishment for something she did. This is why the power in Ryoko's gems are the same as that of Jurian trees which derive their power from Washu's sister, Tsunami. I suspect it had to do with treachery masterminded by Tokimi.
The whole reason that Washu created Ryoko in the first place was so that she could harness the powers within her gems. With all three gems in place Ryoko can use the power that washu once held. Including 10 Lighthawk wings. During Ryoko's creation, Washu's assistants Kagato and Yakage each exploited Washu's creation of Ryoko. Yakage stole some cells and made a clone. Kagato just imprisoned Washu and brainwashed Ryoko.
For centuries the galaxy's most brilliant scientists have researched the gems but they reveal little about their origin or how they work. The closest replication of the gems was achieved by Kagato. He managed to create gems more powerful than Ryoko's but they were not nearly as compact. They were so large that he had them mounted on his ship the Soja (which Washu created.) When Ryoko was under the control of Kagato she had all three at once. But after her battle with Yosho, she was stripped of all three.
Ryoko can create lesser versions of the gems. They each take about 750 years to create and are very fragile. This was the one that Tenchi destroyed in OAV 1. Ryoko wears an earring on her right ear, this is one of her created gems. She used its power in OAV 6 when she fought Kagato. Ryoko's gems sometimes appear in the TV series but they're only for decoration.
There is also a brief appearance of another possible gem which appears when Yosho cuts off of Ryoko's wrist in OAV 5. But it was probably the ontaht Tenchi gave her and it was recovered by Kagato when he retraeated to Soja.
One major drawback, is that Kagato used the gems to control Ryoko and that's why he was able to control her. Although she did possess all three gems when she fought Yosho on earth, she was fighting Kagato's control with all her power which is how Yosho was able to take the gems from her.