Isolate the square root term by multiplying both sides by (n)/(2m).

Step 1: Isolate the square root term by multiplying both sides by $\frac{n}{2m}$. $$T = \frac{2m}{n} \sqrt{\frac{L - A^2}{3k}}$$ $$\frac{Tn}{2m} = \sqrt{\frac{L - A^2}{3k}}$$ Step 2: Eliminate the square root by squaring both sides of the equation. $$\left(\frac{Tn}{2m}\right)^2 = \left(\sqrt{\frac{L - A^2}{3k}}\right)^2$$ $$\frac{T^2 n^2}{4m^2} = \frac{L - A^2}{3k}$$ Step 3: Multiply both sides by $3k$ to isolate the term $(L - A^2)$. $$3k \left(\frac{T^2 n^2}{4m^2}\right) = L - A^2$$ $$\frac{3kT^2 n^2}{4m^2} = L - A^2$$ Step 4: Rearrange the equation to make $A^2$ the subject. $$A^2 = L - \frac{3kT^2 n^2}{4m^2}$$ Step 5: Take the square root of both sides to solve for $A$. $$A = \pm \sqrt{L - \frac{3kT^2 n^2}{4m^2}}$$ The final answer is $\boxed{A = \pm \sqrt{L - \frac{3kT^2 n^2}{4m^2}}}$. That's 2 down. 3 left today — send the next one.