How does 256* sqrt 2 = sqrt 2 to the 17th power?????

Since 256  =  2⁸  and  sqrt(2)  =  2^1/2:

   256·sqrt(2)  =  2⁸·2^1/2  =  2^{8 + 1/2}  =  2^17/2  (when multiplying numbers with the same base, add                                                                                  the exponents)

sqrt(2)¹⁷  =  (2^1/2)¹⁷  =  2^17/2

 

Therefore:  256·sqrt(2)  =  sqrt(2)¹⁷  because both equal  2^17/2.