Ex.1.5 Q2 Number System Solution - NCERT Maths Class 9

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) \((5 - \sqrt 2 )\,(5 + \sqrt 2 )\)

Text Solution

Steps:

(i) \(\begin{align}(3 + \sqrt 3 )\,(2 + \sqrt 2 )\end{align}\)

By Distributive property

\[\begin{align}=6+3 \sqrt{2}+2 \sqrt{3}+\sqrt{6} \end{align}\]

(ii) \(\begin{align}(3 + \sqrt 3 )\,(3 - \sqrt 3 )\end{align}\)

Using the identity:

\[\begin{align}(a+b)(a-b) &=a^{2}-b^{2} \\(3+\sqrt{3}) 3-\sqrt{3} &=9-3 \\ &=6 \end{align}\]

(iii) \(\begin{align}{(\sqrt 5 + \sqrt 2 )^2}\,\end{align}\)

Steps:

Using the identity: \((a+b)^{2} =a^{2}+2 a b+b^{2}\)

\[\begin{align}&=(\sqrt{5})^{2}+2 \times \sqrt{5} \times \sqrt{2}+(\sqrt{2})^{2} \\ &=(5+2 \sqrt{10}+2) \\ &=7+2 \sqrt{10} \end{align}\]

(iv) \(\begin{align}(5 - \sqrt 2 )\,(5 + \sqrt 2 )\end{align}\)

Steps:

Using the identity: \((a+b)(a-b) =a^{2}-b^{2}\)

\[\begin{align}&(\sqrt{5})^{2}-(\sqrt{2})^{2} \\ &=5-2 \\ &=3 \end{align}\]