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  4. The product (32)(32)1/6(32)1/36..... to ∞ is:

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Q. The product $ (32){{(32)}^{1/6}}{{(32)}^{1/36}}..... $ to $ \infty $ is:

KEAMKEAM 2005

Solution:

$ (32){{(32)}^{1/6}}{{(32)}^{(1/36)}}.... $
$ ={{32}^{1+\frac{1}{6}+\frac{1}{36}+.....}} $
$ ={{32}^{\frac{1}{1-1/6}}}={{(32)}^{6/5}} $
$ ={{({{2}^{5}})}^{6/5}}={{2}^{6}}=64 $