Question

In the following equations, find which variables x, y, z etc, represent rational or irrational numbers :

(i) $x^2$ = 5 (ii) $y^2$= 9

(iii) $z^2$ = 0.04 (iv) $u^2=\frac{17}{4}$

(v) $v^2$ = 3 (vi) $w^2$= 27

(vii) $t^2=0.4$

Answer 1

(i) $x^2$ = 5 $\Rightarrow$ $x^2$ = $\pm \sqrt{5}{)}^2$

$\Rightarrow$ x = $\sqrt{5}$

Which is an irrational number.

(ii) $y^2$= 9 $\Rightarrow$ $y^2$ = (3) ${}^2$

$\therefore$ y = 3

(iii) $z^2$ = 0.04 (0.2)${}^2$

$\therefore$ z = 0.2

Which is a rational number.

(iv) $u^2=\frac{17}{4}={\left(\frac{\sqrt{17}}{2}\right)}^2\therefore u=\frac{\sqrt{17}}{2}$

Which is an irrational number.

(v) $v^2=3=(\sqrt{3}{)}^2$

Which is an irrational number.

(vi) $w^2=279\times 3=(3\sqrt{3}{)}^2$

$\therefore$ w = 3$\sqrt{3}$

Which is an irrational number.

(vii) $t^2=0.4\Rightarrow t^2=\frac{4}{10}={\left(\frac{2}{\sqrt{10}}\right)}^2\therefore =\frac{2}{\sqrt{10}}=\frac{2\times \sqrt{10}}{\sqrt{10}\times \sqrt{10}}=\frac{2\sqrt{10}}{10}=\frac{\sqrt{10}}{5}$