24. Solve for \( x: \sqrt{(x-2)^{-3}}=64 \)

Solve the equation \( \sqrt{(x-2)^{-3}}=64 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\sqrt{\left(x-2\right)^{-3}}=64\) - step1: Find the domain: \(\sqrt{\left(x-2\right)^{-3}}=64,x>2\) - step2: Simplify the root: \(\frac{1}{\sqrt{\left(x-2\right)^{3}}}=64\) - step3: Cross multiply: \(1=\sqrt{\left(x-2\right)^{3}}\times 64\) - step4: Simplify the equation: \(1=64\sqrt{\left(x-2\right)^{3}}\) - step5: Swap the sides: \(64\sqrt{\left(x-2\right)^{3}}=1\) - step6: Divide both sides: \(\frac{64\sqrt{\left(x-2\right)^{3}}}{64}=\frac{1}{64}\) - step7: Divide the numbers: \(\sqrt{\left(x-2\right)^{3}}=\frac{1}{64}\) - step8: Raise both sides to the \(2\)-th power\(:\) \(\left(\sqrt{\left(x-2\right)^{3}}\right)^{2}=\left(\frac{1}{64}\right)^{2}\) - step9: Evaluate the power: \(\left(x-2\right)^{3}=\frac{1}{4096}\) - step10: Simplify the equation: \(\sqrt[3]{\left(x-2\right)^{3}}=\sqrt[3]{\frac{1}{4096}}\) - step11: Calculate: \(x-2=\sqrt[3]{\frac{1}{4096}}\) - step12: Simplify the root: \(x-2=\frac{1}{16}\) - step13: Move the constant to the right side: \(x=\frac{1}{16}+2\) - step14: Add the numbers: \(x=\frac{33}{16}\) - step15: Check if the solution is in the defined range: \(x=\frac{33}{16},x>2\) - step16: Find the intersection: \(x=\frac{33}{16}\) - step17: Check the solution: \(x=\frac{33}{16}\) The solution to the equation \( \sqrt{(x-2)^{-3}}=64 \) is \( x = \frac{33}{16} \) or \( x = 2.0625 \).