Show that x is rational, if :

(i) x² = 16

(ii) x² = 0.0004

(iii) x² = $1\dfrac{7}{9}$

Find the value of :

$\dfrac{1}{1 + \sqrt{2}} + \dfrac{1}{\sqrt{2} + \sqrt{3}} + …+ \dfrac{1}{\sqrt{2025} + \sqrt{2026}}$

(i) Real numbers are numbers which include both rational and irrational numbers.

(ii) Every rational number can be expressed as $\dfrac{p}{q}$, where p and q are integers and q ≠ 0.

(iii) Irrational numbers cannot be expressed as $\dfrac{p}{q}$, where p and q are integers and q ≠ 0.

Answer the following questions :

(a) Is the difference between a rational number and an irrational number always rational?

(b) Is 0.5555 ……. a rational number ?

(c) Is 0.505005 …… a rational number ?

(d) Is the product of irrational number (3 - $\sqrt{7}$) and its rationalising factor a rational number?

According to Taruner Swapna Scheme of West Bengal Government, backward / economically disadvantaged students of West Bengal can receive financial aid and devices like Smartphones / Tablets for digital learning, bridging the digital gap and supporting online studies.

Rohan who resides in Purulia district of West Bengal received a tablet and he decided to complete his education through e-learning. One day he was studying number system, where he learnt about rational and irrational numbers, etc. He was particularly interested in irrational numbers like 7 + $4\sqrt{3}$.

Based on the above information, answer the following :

(i) What is the rationalising factor of 7 + $4\sqrt{3}$ ?

(ii) What is the reciprocal of 7 + $4\sqrt{3}$ ? Are rationalising factor and reciprocal same?

(iii) If x = 7 + $4\sqrt{3}$, then find the value of $\Big(x + \dfrac{1}{x}\Big)$.