Number Systems Test - 29

$$7\sqrt{48}-\sqrt{72}-\sqrt{27}+3\sqrt{2}$$
$$=7\times 4\sqrt{3}-6\sqrt{2}-3\sqrt{3}+3\sqrt{2}$$
$$=(28-3)\sqrt{3}+(3-6)\sqrt{2}$$
$$=25 \sqrt{3}-3 \sqrt{2}$$

$$\displaystyle 4\sqrt{12}-\sqrt{75}-7\sqrt{48}$$
$$=4\sqrt{12}-\sqrt{75}-7\sqrt{48}$$
$$=4\times 2 \sqrt{3} - 5 \sqrt{3} -7 \times 4\sqrt{3}$$
$$=\sqrt{3}[8-5-28]$$
$$=-25 \sqrt{3}$$

$${\textbf{Step 1: Define Irrational number.}}$$

               $${\text{An irrational number is a number that cannot be expressed as a dfration}}{\text{.}}$$ 

               $${\text{Also, they have decimal expression neither terminate nor become periodic}}{\text{.}}$$

               $${\text{Now, check it for all the four options}}{\text{.}}$$

$${\textbf{Step 2: Consider, option A.}}$$    

               $$\dfrac{22}{7}$$ $$=3.\overset{\centerdot }{\mathop{1}}\,4285\overset{\centerdot }{\mathop{7}}\,$$

               $$\text{Where }\centerdot \text{ represent beginning and end of recurring group of numbers}\text{.}$$

               $$ \Rightarrow \dfrac{{22}}{7}$$ $${\text{is a rational number as as it is a recurring decimal}}{\text{.}}$$

$${\textbf{Step 3: Consider, option B.}}$$

               $$3.141592$$

               $${\text{It is a terminating decimal number}}{\text{.}}$$

               $$ \Rightarrow {\text{3}}{\text{.141592 is a rational number}}{\text{.}}$$

$${\textbf{Step 4: Consider, option C.}}$$

               $$2.78181818$$

               $${\text{It is a terminating decimal number}}{\text{.}}$$

               $$ \Rightarrow 2.78181818{\text{ is a rational number}}{\text{.}}$$

$${\textbf{Step 5: Consider, option D.}}$$

               $$0.123223222322223 \ldots $$

               $${\text{It is a non - terminating decimal number and also it is not a recurring decimal}}{\text{.}}$$

               $$ \Rightarrow 0.123223222322223 \ldots {\text{ is a irrational number}}{\text{.}}$$

$${\textbf{Final Answer: Hence, option (D)}}$$ $$\mathbf{0.123223222322223 \dots} {\textbf{ is correct answer.}}$$  

The conjugate of $$\sqrt{5}+1$$ is $$\sqrt{5}-1$$
Product = $$(\sqrt{5}+1) \times (\sqrt{5}-1)={\sqrt{5}}^2-1^2=5-1=4$$

Consider option $$A$$.

Let be the Number are $$4\sqrt{3}$$  and  $$2\sqrt{3}$$.
Difference of Number  $$4\sqrt{3} - 2\sqrt{3} = 2\sqrt{3}$$, which is a irrational number.

$$\sqrt { 2 } =1.4142135623730951....$$