Question

if A and B are natural number such that (1/A) ^1 /B =0.3bar then value of AB is ​

Answer 1

it is given that $\left(\frac{1}{A}\right)^{\frac{1}{B}}=0.\bar{3}$

we know, $0.\bar{3}$ can be written as rational number , P/Q , where Q ≠ 0

$0.\bar{3}=\frac{3-0}{9}=\frac{1}{3}$

now, $\left(\frac{1}{A}\right)^{\frac{1}{B}}=\frac{1}{3}$

because A and B are natural numbers so, we can assume A and B in such a way that value of A and B follow above expression.

if we assume A = 3 and B = 1

then, $\left(\frac{1}{3}\right)^{\frac{1}{1}}=\frac{1}{3}$

so, A = 3 and B = 1

then, AB = 3 × 1 = 3

hence, answer is 3.

Answer 2

Answer:

Value of AB is $B3^B$

Step-by-step explanation:

Given:

$(\frac{1}{A})^\frac{1}{B}=\frac{1}{3}$

Taking logarithm on both sides

$log(\frac{1}{A})^\frac{1}{B}=log\frac{1}{3}$

$\frac{1}{B}log(\frac{1}{A})=log1-log3$

$\frac{1}{B}log(\frac{1}{A})=0-log3$

$\frac{1}{B}log(\frac{1}{A})=-log3$

$log(\frac{1}{A})=-Blog3$

$log(\frac{1}{A})=log3^{-B}$

$log(\frac{1}{A})=log(\frac{1}{3^{B}})$

This implies

$\frac{1}{A}=\frac{1}{3^B}$

$A=3^B$

Then,

$AB=(3^B)B$

$AB=B3^B$

when B=1, AB = 3

when B=2, AB = 18

when B=3, AB = 81

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