The correct spelling of the phrase "study at approve" is "study and prove." The first word, "study," is pronounced /ˈstʌdi/, with emphasis on the first syllable and a short 'u' sound. The second word, "and," is pronounced /ænd/, with a short 'a' and a soft 'd' sound. The third word, "prove," is pronounced /pruːv/, with a long 'u' sound and emphasis on the second syllable. When spelling this phrase, it is important to use the correct combination of letters to accurately represent each sound in the word.