Exponents and Surds

Simplify: $\frac{\sqrt{18}}{\sqrt{2}}$

The value of $x^3+\frac{1}{x^3}$ if $x=\sqrt{3}+\sqrt{2}$ is $a\sqrt{b}$. Find $a+b$

If $2\mathrm{r}=\mathrm{h}+\sqrt{{\mathrm{r}}^2+{\mathrm{h}}^2}$, then what is the value of $\mathrm{r} : \mathrm{h}$? ($\mathrm{r}\ne 0$)

Find the value of $x^3-\frac{1}{x^3}$ if $x=\sqrt{5}+2$.

If $\mathrm{m}=\sqrt{\frac{\mathrm{n}}{\mathrm{n}+{\displaystyle \frac{1}{2}}}}$ and $\mathrm{m}=\frac{1}{2}$, then the value of $\mathrm{n}$ is _____.

Express $y$ in terms of $x$ if $x=y+\sqrt{y^2+1}$.

If ${\left(\frac{2}{3}\right)}^6\times {\left(\frac{9}{4}\right)}^5={\left(\frac{3}{2}\right)}^{\mathrm{m}+2}$, then find the value of $\mathrm{m}$.

If $2^x\times 3^y\times 5^z=2160,$ find $x, y$ and $z.$ Hence, compute the value (as an improper fraction) of  $3^x\times 2^{-y}\times 5^{-z}.$

Is the product of the square root of $x$ with the cube root of $x$, is sixth root of the fifth power of $x$. (Yes $/$ No)

Write ${\left(\frac{1}{9}\right)}^{\frac{-1}{2}}\times {\left(64\right)}^{\frac{-1}{3}}$ as a rational number.

If $a$ and $b$ are distinct positive primes such that $\sqrt[3]{a^6{b}^{-4}}=a^x{b}^{2y}$, then find the value of $x+y$, as an improper fraction.

Solve $\sqrt{\frac{a}{b}}={\left(\frac{b}{a}\right)}^{1-2x},$ where $a, b$ are distinct positive primes.

If $5^{3x}=125$ and ${10}^y=0.001$, then find the value of $x+y$.

If ${27}^x=\frac{9}{3^x},$ then find the value of $x.$

Assuming that $x$ is a positive real numbers, if the simplified form of ${\left(\sqrt{x^{-3}}\right)}^5$ is $\frac{1}{x^{\frac{m}{n}}}$, then find the value of $m+n$.

Find the value of $x$ [in $\frac{p}{q}$ form] :

$7^{2x+3}=1$

If the hypotenuse of the fourth triangle is $\sqrt{k}$ then find $k$.

In the picture, the square on the hypotenuse of the topmost right triangle is drawn. If the side of the square is $k {\mathrm{m}}^2$ then find $k$. 

If the hypotenuse of the right triangle with perpendicular sides $\sqrt{2} \mathrm{cm}$ and $\sqrt{3} \mathrm{cm}$ is $\sqrt{k} \mathrm{cm}$, then find the value of $k$.

We have seen how we can draw a series of right triangles as in the picture. The lengths of the sides of the tenth triangle drawn like this are