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Chapter 10: Q49. (page 528)

Solve the equation ${(\frac{1}{9})}^{m} = 81^{m + 4}$ . Check your solution.

Short Answer

Expert verified

The solution for the equation is $m = - \frac{8}{3}$ .

Step by step solution

01

Step 1. Write down the given equation.

The given equation is ${(\frac{1}{9})}^{m} = 81^{m + 4}$ .

02

Step 2. Simplify the equation.

The given equation ${(\frac{1}{9})}^{m} = 81^{m + 4}$ is solved as:

$\begin{array}{l} {(\frac{1}{9})}^{m} = 81^{m + 4} .... (Given) \\ {(9)}^{鈭� m} = {(9^{2})}^{m + 4} \\ {(9)}^{鈭� m} = {(9)}^{2 m + 8} \\ 鈭� m = 2 m + 8 \\ 3 m = 鈭� 8 \\ m = 鈭� \frac{8}{3} \end{array}$

03

Step 3. Verify the result for obtained value.

Plugging $m = - \frac{8}{3}$ in given equation and verify the result.

$\begin{array}{l} {(\frac{1}{9})}^{m} = 81^{m + 4} .... (Given) \\ {(\frac{1}{9})}^{(鈭� \frac{8}{3})} = 81^{(鈭� \frac{8}{3}) + 4} \\ {(9)}^{(\frac{8}{3})} = {(9^{2})}^{(\frac{4}{3})} \\ 9^{\frac{8}{3}} = 9^{\frac{8}{3}} .... (True) \end{array}$

Hence the solution is verified for $m = - \frac{8}{3}$ .

04

Step 4. Conclusion.

The solution for the given equation is $m = - \frac{8}{3}$ .

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