S i m p l i f y space semicolon space square root of m squared n squared space end root X space space root index 6 of m squared n squared end root space x space cube root of m squared n squared end root

Asked by kishor.kp14 | 21st Jun, 2019, 01:58: PM

Expert Answer:

begin mathsize 16px style square root of straight m squared straight n squared end root cross times root index 6 of straight m squared straight n squared end root cross times cube root of straight m squared straight n squared end root
equals space open parentheses straight m squared straight n squared close parentheses to the power of 1 half end exponent space cross times space open parentheses straight m squared straight n squared close parentheses to the power of 1 over 6 end exponent space cross times open parentheses straight m squared straight n squared close parentheses to the power of 1 third end exponent space
equals open parentheses mn close parentheses to the power of 2 cross times 1 half space space space end exponent cross times space open parentheses mn close parentheses to the power of 2 cross times 1 over 6 space space space end exponent cross times space open parentheses mn close parentheses to the power of 2 cross times 1 third space space space end exponent space space space space space space space space space space space space.... space since space open parentheses straight x to the power of straight a close parentheses to the power of straight b equals space straight x to the power of ab
equals open parentheses mn close parentheses space cross times space open parentheses mn close parentheses to the power of 1 third space space space end exponent cross times space open parentheses mn close parentheses to the power of 2 over 3 space space space end exponent space space space space space space space space space space space space space space space space space space space space space space space space
equals open parentheses mn close parentheses to the power of 1 space plus space 1 third end exponent space cross times space open parentheses mn close parentheses to the power of 2 over 3 space space space end exponent space space space space space space space space space space space space space space space space space.... space since space straight x to the power of straight a cross times space straight x to the power of straight b space equals space straight x space to the power of straight a space plus space straight b end exponent
equals open parentheses mn close parentheses to the power of 1 space plus space 1 third plus space 2 over 3 end exponent space space
equals open parentheses mn close parentheses to the power of fraction numerator 3 space plus space 1 space plus space 2 over denominator 3 end fraction end exponent
equals open parentheses mn close parentheses to the power of 6 over 3 end exponent
equals open parentheses mn close parentheses squared
equals straight m squared straight n squared
end style

Answered by Yasmeen Khan | 21st Jun, 2019, 04:45: PM