Insert a rational number and an irrational number between 5 and 6.

Insert two rational numbers and two irrational numbers between ${\sqrt3}$ and ${\sqrt8}$.

State which of following real numbers are :

-8, 0, ${\sqrt5}$, $\dfrac{5}{7}$, $-{\sqrt{18}}$, ${\sqrt{32}}$, 4.28, π, 3, $-\dfrac{8}{15}$, 0.075

(i) rational

(ii) irrational

(iii) positive integers

(iv) negative integers

(v) neither positive nor negative.

Examine whether the following numbers are rational or irrational :

(i) $(3- \sqrt5)^2$

(ii) $(7 - \sqrt7)(7 + \sqrt7)$

(iii) $(2\sqrt3 + 3\sqrt2)^2$

(iv) $(2\sqrt3 - 3\sqrt2)(2\sqrt3 + 3\sqrt2)$

(v) $5\sqrt3 \times \sqrt{12}$