Question

If $a$ and $b$ are natural numbers such that ${\left(\frac{1}{a}\right)}^{\frac{1}{b}}=0.\overline{3},$ then the value of $ab$ is:

• A. $81$
• B. $24$
• C. $192$
• D. $375$

Answer 1

The correct option is A

$81$

Step1: Simplify the given expression

Given $a$ and $b$ are natural numbers such that ${\left(\frac{1}{a}\right)}^{\frac{1}{b}}=0.\overline{3}$

Now $$\begin{gathered} {\left(\frac{1}{a}\right)}^{\frac{1}{b}}=0.\overline{3} \\ \Rightarrow {\left(\frac{1}{a}\right)}^{\frac{1}{b}}=\frac{1}{3} \end{gathered}$$

Step 2: Make Possible combinations to match correct options

$$\begin{gathered} When,{\left(\frac{1}{3}\right)}^{\frac{1}{1}}=\frac{1}{3} \\ \Rightarrow a=3,b=1 \\ \therefore a.b=3 \\ When,{\left(\frac{1}{9}\right)}^{\frac{1}{2}}=\frac{1}{3} \\ \Rightarrow a=9,b=2 \\ \therefore a.b=18 \\ When,{\left(\frac{1}{27}\right)}^{\frac{1}{3}}=\frac{1}{3} \\ \Rightarrow a=27,b=3 \\ \therefore a.b=81 \\ when,{\left(\frac{1}{81}\right)}^{\frac{1}{4}}=\frac{1}{3} \\ \Rightarrow a=81,b=4 \\ \therefore a.b=324 \\ \therefore Outofavailableoptions \\ Possibleanswerisa.b=81 \end{gathered}$$

Final answer:

Hence, the correct option is A.