Question:
If m and n are integers such that $(\sqrt{2})^{19} 3^{4} 4^{2} 9^{m} 8^{n}=3^{n} 16^{m}(\sqrt[4]{64})$ then m is
- $-20$
- $-12$
- $-24$
- $-16$
Correct Answer: Option: 2
$(\sqrt{2})^{19} 3^{4} 4^{2} 9^{m} 8^{n}=3^{n} 16^{m}(\sqrt[4]{64})$
$\Rightarrow 2^{19 / 2} \times 3^{4} \times 2^{4} \times 3^{2 m} \times 2^{3 n}=3^{n} \times 2^{4 m} \times 2^{3 / 2}$
$\Rightarrow {{2}^{(19/2+4+3n)}}\times {{3}^{(4+2m)}}={{2}^{(4m+3)}}\times {{3}^{n}}$
Comparing the powers of same bases we get
$\frac{19}{2}+4+3 n=4 m+\frac{3}{2} \cdots(1)$
$4+2 m=n \cdots(2)$
Substitute the value of n from (2) in (1) and solving for m, we get m = -12
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