Question

If 2a =5b=10c, prove that 1/a+1/b=1/c

Answer 1

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$2^{a} = 5^{b} = 10^{c} \: (given )$

$\red{ To \:show : \frac{1}{a} + \frac{1}{b} = \frac{1}{c} }$

$\green{Solution: }$

$Let \: 2^{a} = 5^{b} = 10^{c} = k$

$i) 2^{a} = k \: \implies 2 = k^{\frac{1}{a}} \: --(1)$

$ii) 5^{b} = k \: \implies 5 = k^{\frac{1}{b}} \: --(2)$

$i) 10^{c} = k \\ \implies 10 = k^{\frac{1}{c}}$

$\implies 2 \times 5 = k^{\frac{1}{c}}$

$\implies k^{\frac{1}{a}} \times k^{\frac{1}{b}}= k^{\frac{1}{c}}$

$\implies k^{\frac{1}{a} + \frac{1}{b}} = k^{\frac{1}{c}}$

$\boxed { \pink { Since, a^{m} \times a^{n} = a^{m+n} }}$

$\implies \frac{1}{a} + \frac{1}{b}= \frac{1}{c}$

$\boxed { \blue { Since, a^{m} = a^{n} \implies m = n }}$

$Hence , proved$

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Answer 2

Answer:

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