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Please help!

1- It is a four foot five inches exterior sign or it is a four feet five inches exterior sign.
2- It is a 200 square foot kitchen or It is a 200 square feet kitchen.
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In both cases "foot" would be normal usage. There is no reason why you cannot use either foot or feet.
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Feebs11In both cases "foot" would be normal usage. There is no reason why you cannot use either foot or feet.
And 'inch' in the singular as well.
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Comments  
In the adjective position, such expressions are always singular.

No one ever says a five-dollars bill, for example.

CJ
Students: Are you brave enough to let our tutors analyse your pronunciation?
You can use eight-foot boards to side a house, but “foot” is correct only in this sort of adjectival phrase combined with a number (and usually hyphenated). The boards are eight feet (not foot) long. It’s always X feet per second and X feet away.

SOURCE: http://wsu.edu/~brians/errors/foot.html