M L Aggarwal Solutions for Chapter: Number System, Exercise 4: Exercise 1.4

If $x=1-\sqrt{2}$, find the value of ${\left(x-\frac{1}{x}\right)}^4$.

If $x=7+4\sqrt{3}$, then find the value of $\sqrt{x}+\frac{1}{\sqrt{x}}$.

Hint: $x=7+4\sqrt{3}=4+3+4\sqrt{3}={\left(2+\sqrt{3}\right)}^2$

       $\Rightarrow \sqrt{x}=2+\sqrt{3}$

If $x=5-2\sqrt{6}$, find the value of $x^2+\frac{1}{x^2}$.

If $p=\frac{2-\sqrt{5}}{2+\sqrt{5}}$ and $q=\frac{2+\sqrt{5}}{2-\sqrt{5}}$, find the value of $p+q$.

If $p=\frac{2-\sqrt{5}}{2+\sqrt{5}}$ and $q=\frac{2+\sqrt{5}}{2-\sqrt{5}}$, find the value of $p-q$.

If $p=\frac{2-\sqrt{5}}{2+\sqrt{5}}$ and $q=\frac{2+\sqrt{5}}{2-\sqrt{5}}$, find the value of $p^2+q^2$.

If $p=\frac{2-\sqrt{5}}{2+\sqrt{5}}$ and $q=\frac{2+\sqrt{5}}{2-\sqrt{5}}$, find the value of $p^2-q^2$.

If $x=\frac{\sqrt{2}-1}{\sqrt{2}+1}$ and $y=\frac{\sqrt{2}+1}{\sqrt{2}-1}$, then find the value of $x^2+5xy+y^2$.

Hint: $x^2+5xy+y^2={\left(x+y\right)}^2+3xy$