Question

if 2 to the power a is equal to 3 to the power b is equal to 6 to the power C then prove that C equals to a b by a + b​

Answer 1

Answer:

$\mathsf{c=\frac{ab}{a+b}}}$

Step-by-step explanation:

$\textsf{Given:}$

$\mathf{2^a=3^b=6^c=k(say)}$

$\mathf{2^a=k,\;\;\;3^b=k,\;\;\;6^c=k}$

$\mathf{\implies\;2=k^{\frac{1}{a}},\;\;\;3=2=k^{\frac{1}{b}},\;\;\;6=k^{\frac{1}{c}}}$

Now,

$\mathsf{6=k^{\frac{1}{c}}}$

$\mathsf{\implies\;2{\times}3=k^{\frac{1}{c}}}$

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

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

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

$\textsf{Equating powers on both sides, we get}$

$\mathsf{\frac{a+b}{ab}=\frac{1}{c}}$

$\textsf{Taking reciprocals on both sides, we get}$

$\mathsf{\frac{ab}{a+b}=c}$

$\implies\boxed{\mathsf{c=\frac{ab}{a+b}}}$

Answer 2

Step-by-step explanation:

mathsf{c=\frac{ab}{a+b}}}

Step-by-step explanation:

\textsf{Given:}Given:

\mathf{2^a=3^b=6^c=k(say)}\mathf2

a

=3

b

=6

c

=k(say)

\mathf{2^a=k,\;\;\;3^b=k,\;\;\;6^c=k}\mathf2

a

=k,3

b

=k,6

c

=k

\mathf{\implies\;2=k^{\frac{1}{a}},\;\;\;3=2=k^{\frac{1}{b}},\;\;\;6=k^{\frac{1}{c}}}\mathf⟹2=k

a

1

,3=2=k

b

1

,6=k

c

1

Now,

\mathsf{6=k^{\frac{1}{c}}}6=k

c

1

\mathsf{\implies\;2{\times}3=k^{\frac{1}{c}}}⟹2×3=k

c

1

\mathsf{\implies\;k^{\frac{1}{a}}{\times}k^{\frac{1}{b}}=k^{\frac{1}{c}}}⟹k

a

1

×k

b

1

=k

c

1

\mathsf{\implies\;k^{\frac{1}{a}+\frac{1}{b}}=k^{\frac{1}{c}}}⟹k

a

1

+

b

1

=k

c

1

\mathsf{\implies\;k^{\frac{a+b}{ab}}=k^{\frac{1}{c}}}⟹k

ab

a+b

=k

c

1

\textsf{Equating powers on both sides, we get}Equating powers on both sides, we get

\mathsf{\frac{a+b}{ab}=\frac{1}{c}}

ab

a+b

=

c

1

\textsf{Taking reciprocals on both sides, we get}Taking reciprocals on both sides, we get

\mathsf{\frac{ab}{a+b}=c}

a+b

ab

=c

\implies\boxed{\mathsf{c=\frac{ab}{a+b}}}⟹

c=

a+b

ab