Question

If a^m = b^n = (ab)^mn then m+n is

Answer 1

Answer:

1

Step-by-step explanation:

${a}^{m} = {b}^{n} = {(ab)}^{mn} = k$

$a = {k}^{ \frac{1}{m} } \: \: \: \: b = {k}^{ \frac{1}{n} } \: \: \: \: ab = \: {k}^{ \frac{1}{mn} }$

$ab \: = \: {k}^{ \frac{1}{m} } \times {k}^{ \frac{1}{n} } = {k}^{ \frac{1}{m} + \frac{1}{n} }$

$ab \: = \: ab$

${k}^{ \frac{1}{m} + \frac{1}{n} } \: = \: {k}^{ \frac{1}{mn} }$

Bases are same, So we can equate the powers.

$\frac{1}{m} + \frac{1}{n} \: = \: \frac{1}{mn}$

$\frac{n + m}{mn} = \frac{1}{mn}$

$m + n \: = \: \frac{1 \times mn}{mn}$

$m + n \: = \: 1$

Hope this helps

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