In 1990, a missouri woman placed a sign protesting the persian gulf war in her yard, despite a city ordinance against yard signs. in ladue v. gilleo (1994), the supreme court ruled that the ordinance in the city of laude, missouri prohibiting yard signs violated ms. gilleo's first amendment rights. what power did the supreme court apply in this case? .

In this case, the Supreme Court applied the power to declare a law/an ordinance unconstitutional.

Explanation:

This is what is known as "judicial review", which conveys the idea that the actions performed by the executive and the legislative branches of a government are subject to review and they may be invalidated by the Court.  

This power gives the Supreme Court an active role, since it controls that the two other branches observe the US Constitution. In the case at issue, the Supreme Court set forth that the ordinance violated Ms. Gilleo's First Amendment rights, which include, among others, the right of freedom of speech. Ms. Gilleo had placed a protesting sign, the city issued an ordinance against such sign, against her right of freedom of speech. Therefore, the ordinance the city issued was considered to be unconstitutional and, as a consequence, null and void.

The US Supreme Court has the power to determine whether a law is unconstitutional by applying this constitutional doctrine called "judicial review".  

the fourth amendment protects:

"the right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no warrants shall issue, but upon probable cause, supported by oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized."  

so this would mean the denial to search and seizure without warrant or permission.