From Förberedande kurs i matematik 1

If we write the numbers in power form,

\displaystyle \sqrt{7}=7^{1/2}\quad and \displaystyle \quad 7=7^1\,,

it becomes clear that 7 is larger than \displaystyle \sqrt{7}, because both numbers have the same base, 7, which is larger than 1 and the exponent 1 is greater than 1/2.