the prove that,
1/a=1/b+1/c
Answers
Answer:
/* There is typing errors in the question . It must be like this*/
2^{a} = 5^{b} = 10^{c} \: (given )2
a
=5
b
=10
c
(given)
\red{ To \:show : \frac{1}{a} + \frac{1}{b} = \frac{1}{c} }Toshow:
a
1
+
b
1
=
c
1
\green{Solution: }Solution:
Let \: 2^{a} = 5^{b} = 10^{c} = kLet2
a
=5
b
=10
c
=k
i) 2^{a} = k \: \implies 2 = k^{\frac{1}{a}} \: --(1)i)2
a
=k⟹2=k
a
1
−−(1)
ii) 5^{b} = k \: \implies 5 = k^{\frac{1}{b}} \: --(2)ii)5
b
=k⟹5=k
b
1
−−(2)
\begin{gathered} i) 10^{c} = k \\ \implies 10 = k^{\frac{1}{c}} \end{gathered}
i)10
c
=k
⟹10=k
c
1
\implies 2 \times 5 = k^{\frac{1}{c}}⟹2×5=k
c
1
\implies k^{\frac{1}{a}} \times k^{\frac{1}{b}}= k^{\frac{1}{c}}⟹k
a
1
×k
b
1
=k
c
1
\implies k^{\frac{1}{a} + \frac{1}{b}} = k^{\frac{1}{c}}⟹k
a
1
+
b
1
=k
c
1
\boxed { \pink { Since, a^{m} \times a^{n} = a^{m+n} }}
Since,a
m
×a
n
=a
m+n
\implies \frac{1}{a} + \frac{1}{b}= \frac{1}{c}⟹
a
1
+
b
1
=
c
1
\boxed { \blue { Since, a^{m} = a^{n} \implies m = n }}
Since,a
m
=a
n
⟹m=n
Hence , provedHence,proved
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