Solve T=mgsqrt{1+{left(r/hright)}^2}

$$T = mg \sqrt{ 1+ { \left( \frac{ r }{ h } \right) }^{ 2 } }$$

Solve for g

$$\left\{\begin{matrix}g=\frac{T|h|}{m\sqrt{r^{2}+h^{2}}}\text{, }&h\neq 0\text{ and }m\neq 0\\g\in \mathrm{R}\text{, }&T=0\text{ and }m=0\text{ and }h\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

To raise \frac{r}{h} to a power, raise both numerator and denominator to the power and then divide.

$$T=mg\sqrt{1+\frac{r^{2}}{h^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{h^{2}}{h^{2}}.

$$T=mg\sqrt{\frac{h^{2}}{h^{2}}+\frac{r^{2}}{h^{2}}}$$

Since \frac{h^{2}}{h^{2}} and \frac{r^{2}}{h^{2}} have the same denominator, add them by adding their numerators.

$$T=mg\sqrt{\frac{h^{2}+r^{2}}{h^{2}}}$$

Swap sides so that all variable terms are on the left hand side.

$$mg\sqrt{\frac{h^{2}+r^{2}}{h^{2}}}=T$$

The equation is in standard form.

$$\sqrt{\frac{r^{2}+h^{2}}{h^{2}}}mg=T$$

Divide both sides by m\sqrt{\left(h^{2}+r^{2}\right)h^{-2}}.

$$\frac{\sqrt{\frac{r^{2}+h^{2}}{h^{2}}}mg}{\sqrt{\frac{r^{2}+h^{2}}{h^{2}}}m}=\frac{T}{\sqrt{\frac{r^{2}+h^{2}}{h^{2}}}m}$$

Dividing by m\sqrt{\left(h^{2}+r^{2}\right)h^{-2}} undoes the multiplication by m\sqrt{\left(h^{2}+r^{2}\right)h^{-2}}.

$$g=\frac{T}{\sqrt{\frac{r^{2}+h^{2}}{h^{2}}}m}$$

Divide T by m\sqrt{\left(h^{2}+r^{2}\right)h^{-2}}.

$$g=\frac{T|h|}{m\sqrt{r^{2}+h^{2}}}$$

Solve for T

$$T=\frac{gm\sqrt{r^{2}+h^{2}}}{|h|}$$

$$h\neq 0$$