M L Aggarwal Solutions for Exercise 5: EXERCISE 1.5

simplify:

$\frac{1}{\sqrt{4}+\sqrt{5}}+\frac{1}{\sqrt{5}{\displaystyle +}{\displaystyle \sqrt{6}}}+\frac{1}{\sqrt{6}{\displaystyle +}{\displaystyle \sqrt{7}}}+\frac{1}{\sqrt{7}{\displaystyle +}{\displaystyle \sqrt{8}}}+\frac{1}{\sqrt{8}{\displaystyle +}{\displaystyle \sqrt{9}}}$

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

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

If $\mathrm{p}=\frac{2-\sqrt{5}}{2+\sqrt{5}} \text{and }\mathrm{q}=\frac{2+\sqrt{5}}{2-\sqrt{5}}$, find the values of:

$\text{p+q}$

If $p=\frac{2-\sqrt{5}}{2+\sqrt{5}} \text{and} q=\frac{2+\sqrt{5}}{2-\sqrt{5}}$, find the values of: $p-q$

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

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

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