Show how $\sqrt 5$ can be represented on the number line.

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We know that $\sqrt{4}=2$

Let $OA$ be a line of length $2$ unit on the number line. 

Now, construct $AB$ of unit length perpendicular to $OA$. and join $OB$.

Now, in right angle triangle $OAB$, by Pythagoras theorem

Now,  take $O$ as centre and $OB$ as radius, draw an arc intersecting number line at $C$. 

Point $C$ represent $\sqrt{5} $ on a number line.

1098-s11

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Show that $0.2353535 \ldots=0.2 \overline{35}$ can be expressed in the form $\frac{p}{q},$ where $p$ and $q$ are integers and $q \neq 0$.

Classroom activity (Constructing the 'square root spiral') : Take a large sheet of paper and construct the 'square root spiral' in the following fashion. Start with a point $O$ and draw a line segment $OP_1$ of unit length. Draw a line segment $P_1P_2$ perpendicular to $OP_1$ of unit length (see Fig.). Now draw a line segment $P_2P_3$ perpendicular to $OP_2$. Then draw a line segment $P_3P_4 $ perpendicular to $OP_3$. Continuing in this manner, you can get the line segment $P_{n-1}P_n$ by drawing a line segment of unit length perpendicular to $OP_{n-1}$. In this manner, you will have created the points $P_2$, $P_3$,...., $P_n$,... ., and joined them to create a beautiful spiral depicting $\sqrt 2,\, \sqrt 3, \,\sqrt 4$, ..............

Simplify each of the following expressions :

$(i)$ $(3+\sqrt{3})(2+\sqrt{2})$

$(ii)$ $(3+\sqrt{3})(3-\sqrt{3})$

$(iii)$ $(\sqrt{5}+\sqrt{2})^{2}$

$(iv)$ $(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})$

Simplify the following expressions :

$(i)$ $(5+\sqrt{7})(2+\sqrt{5})$

$(ii)$ $(5+\sqrt{5})(5-\sqrt{5})$

$(iii)$ $(\sqrt{3}+\sqrt{7})^{2}$

$(iv)$ $(\sqrt{11}-\sqrt{7})(\sqrt{11}+\sqrt{7})$

Classify the following numbers as rational or irrational :

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

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

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

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

$(v)$ $2 \pi$