Surds and Indices

Arrange $\sqrt[4]{3} ,\sqrt[6]{10 },\sqrt[12]{25}$ in descending order.

$$2+\cfrac { 6 }{ \sqrt { 3 } } +\cfrac { 1 }{ 2+\sqrt { 3 } } +\cfrac { 1 }{ \sqrt { 3 } -2 } =?$$

Arrange the following in descending order:

$\sqrt[3]{4}, \sqrt{2}, \sqrt[6]{3}, \sqrt[4]{5}$

If $x=1+\sqrt{2}+\sqrt{3}$, then $x+\frac{1}{x-1}$ will be equal to:

Find the value of $\sqrt{732108}$.

Find the value of:

$\sqrt{-\sqrt{3}+\sqrt{3+8\sqrt{7+4\sqrt{3}}}}$

If $\mathrm{n}=7+4\sqrt{3}$, then the value of $\left(\sqrt{n}+\frac{1}{\sqrt{n}}\right)$ is:

The irrational number $\frac{8}{3+\sqrt{n}}$ when rationalized becomes $3-\sqrt{n}$. What is the value of $n$?

If$3^{(x-y)}=27$  and$3^{(x+y)}=243$, then $x$ is equal to

$9\sqrt{x}=\sqrt{12}+\sqrt{147}$ then find $x=?$

$\frac{\sqrt{10\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}}}{\sqrt[3]{8}}=?$

$a=\frac{\sqrt{3}}{2}$ then find $\sqrt{1+a}+\sqrt{1-a}=?$

Find the value $\sqrt{12+\sqrt{12+\sqrt{12\dots }}}$

Find the value of x and y? If $\frac{7+\sqrt{5}}{7-\sqrt{5}}-\frac{7-\sqrt{5}}{7+\sqrt{5}}=x+7\sqrt{5}y$

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

If $a=3+2\sqrt{2}$ and $ax=1$ then find $a^2+ax+x^2=?$

$\frac{1}{3-\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}$ is equal to?

The simplest value of $\frac{3\sqrt{8}-2\sqrt{12}+\sqrt{20}}{3\sqrt{18}-2\sqrt{27}+\sqrt{45}}=?$

If the value of  $\sqrt{30}$ is approximately $5.477$, then what is the approximate value of  $\sqrt{\frac{5}{6}}$?

For any positive real number x, write the value of:

${\left\{\right({\mathrm{x}}^{\mathrm{a}}{)}^{\mathrm{b}}\}}^{\frac{1}{\mathrm{ab}}}{\left\{\right({\mathrm{x}}^{\mathrm{b}}{)}^{\mathrm{c}}\}}^{\frac{1}{\mathrm{bc}}}{\left\{\right({\mathrm{x}}^{\mathrm{a}}{)}^{\mathrm{c}}\}}^{\frac{1}{\mathrm{ac}}}$