Solve the equation for $$ x $$ : $$ 4(x-2)^{-\frac{3}{4}}=\sqrt{2} $$

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Solve the equation $$ 4(x-2)^{-\frac{3}{4}}=\sqrt{2} $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$4\left(x-2ight)^{-\frac{3}{4}}=\sqrt{2}$$
- step1: Find the domain:
  $$4\left(x-2ight)^{-\frac{3}{4}}=\sqrt{2},x>2$$
- step2: Rewrite the expression:
  $$\frac{4}{\left(x-2ight)^{\frac{3}{4}}}=\sqrt{2}$$
- step3: Cross multiply:
  $$4=\left(x-2ight)^{\frac{3}{4}}\sqrt{2}$$
- step4: Simplify the equation:
  $$4=\sqrt{2}	imes \left(x-2ight)^{\frac{3}{4}}$$
- step5: Swap the sides:
  $$\sqrt{2}	imes \left(x-2ight)^{\frac{3}{4}}=4$$
- step6: Divide both sides:
  $$\frac{\sqrt{2}	imes \left(x-2ight)^{\frac{3}{4}}}{\sqrt{2}}=\frac{4}{\sqrt{2}}$$
- step7: Divide the numbers:
  $$\left(x-2ight)^{\frac{3}{4}}=\frac{4}{\sqrt{2}}$$
- step8: Raise both sides to the $$\frac{4}{3}$$-th power$$:$$
  $$\left(\left(x-2ight)^{\frac{3}{4}}ight)^{\frac{4}{3}}=\left(\frac{4}{\sqrt{2}}ight)^{\frac{4}{3}}$$
- step9: Evaluate the power:
  $$x-2=4$$
- step10: Move the constant to the right side:
  $$x=4+2$$
- step11: Add the numbers:
  $$x=6$$
- step12: Check if the solution is in the defined range:
  $$x=6,x>2$$
- step13: Find the intersection:
  $$x=6$$
- step14: Check the solution:
  $$x=6$$
The solution to the equation $$4(x-2)^{-\frac{3}{4}}=\sqrt{2}$$ is $$x=6$$.
$$ x=6 $$

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