Question

Which of the given numbers is/are rational?
(i) $\sqrt{{\pi }^2}$ (ii) ${}^3\sqrt{0.8}$ (iii) ${}^4\sqrt{0.00016}$ (iv) ${}^3\sqrt{-1}\times \sqrt{{\left(0.09\right)}^{-1}}$

• A. (i) and (iv)
• B. (i) only
• C. (iv) only
• D. All of the given

Answer 1

The correct option is C (iv) only

(i) We know that ${\pi }^2=10$, therefore its equal to, $\sqrt{10}$ and its known that its not rational.

(ii) Lets write it as $\sqrt[3]{30.8}=\sqrt[3]{3\frac{8}{10}}=\frac{2}{\sqrt[3]{310}}$, which is irrational.

(iii) Lets write the given value in scientific notation, $\sqrt[4]{40.00016}=\sqrt[4]{416\times {10}^{-5}}=2\times {10}^{-1}\times \sqrt[4]{40.1}=0.2\times \sqrt[4]{40.1}$, which is clearly irrational.

(iv) Lets rewrite this in scientific notation, $(-1{)}^{1/3}\times (9\times {10}^{-2}{)}^{-1/2}=-1\times 9^{-1/2}\times 10=-1\times \frac{1}{3}\times 10=-\frac{10}{3}$ which is rational.