question_answer

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