Find six rational numbers between $3$ and $4$.

Look at several examples of rational numbers in the form $\frac{p}{q}(q \neq 0),$ where $p$ and $q$ are integers with no common factors other than $1$ and having terminating decimal representations (expansions). Can you guess what property $q$ must satisfy ?

Visualise $4. \overline{26}$ . on the number line, up to $4$ decimal places.

Classify the following numbers as rational or irrational :

$(i)$ $2-\sqrt{5}$

$(ii)$ $(3+\sqrt{23})-\sqrt{23}$

$(iii)$ $\frac{2 \sqrt{7}}{7 \sqrt{7}}$

$(iv)$ $\frac{1}{\sqrt{2}}$

$(v)$ $2 \pi$

Simplify

$(i)$ $2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}}$

$(ii)$ $\left(3^{\frac{1}{5}}\right)^{4}$

$(iii)$ $\frac{7^{\frac{1}{5}}}{7^{\frac{1}{3}}}$

$(iv)$ $13^{\frac{1}{5}} \cdot 17^{\frac{1}{5}}$

Is zero a rational number ? Can you write it in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \ne 0$ ?