Find the values of n, when: |
(a) \[{{5}^{2n}}\times {{5}^{3}}={{5}^{9}}\] |
(b) \[8\times {{2}^{n+2}}=32\]. |
Answer:
(a) \[\begin{align} & {{52}^{2n}}\times {{5}^{3}}={{5}^{9}} \\ & \,\,\,\,\,\,\,\,\,\,{{5}^{2n+3}}={{5}^{9}} \\ \end{align}\] As base 5 is same on both sides \[\therefore \] 2n + 3 = 9 2n = 9 - 3 2n =6 Thus, n = \[\frac{6}{2}=3\] (b) \[8\times {{2}^{n+2}}=32\] \[2\times 2\times 2\times {{2}^{n+2}}=\text{ }2\times 2\times 2\times 2\times 2\] \[{{2}^{3}}\times {{2}^{n+2}}={{2}^{5}}\] \[{{2}^{n+2+3}}={{2}^{5}}\] \[{{2}^{n+5}}={{2}^{5}}\] As base is same on both sides \[\therefore \] n + 5 = 5 n = 5 - 5 = 0
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