Questions based on Operations of Irrational Numbers

Q. The product of adding, subtracting, multiplying, or dividing two irrational numbers is always irrational.

1. True
2. False

Q. $\text{The value of the expression}{p}^2-q^2,\text{if}p=(5+2\sqrt{3})\&q=(5-2\sqrt{3})\text{is}$

1. $40\sqrt{3}$
2. $-40\sqrt{3}$
3. $40$.
4. $-40$

Q. If $19-6\sqrt{10}=a^2,$ find the value of $a.$

1. $\sqrt{10}-3$
2. $\sqrt{3}-10$
3. $\sqrt{10}+3$
4. Cannot be determined.

Q. Find the possible value(s) of the expression: $\sqrt{2+\sqrt{2+\sqrt{2+...}}}$

1. $1$
2. $-1$
3. $2$
4. $-2$

Q. Given that $x=6-4\sqrt{2}$ and $y$ is conjugate of $x,$ the value of $\sqrt{4xy}$ is .

Q. In the expression $y^{(\sqrt{2}+\sqrt{6})}=\frac{1}{\sqrt[4]{4{11}^{(\sqrt{2}-\sqrt{6})}}},$ find the value of $y.$

1. $y={11}^{\frac{2-\sqrt{3}}{4}}$
2. $y={11}^{\frac{4-\sqrt{3}}{4}}$
3. $y={11}^{\frac{2+\sqrt{3}}{4}}$