[(sqrt [4](x)-sqrt [4](a))^-1+(sqrt [4](x)+sqrt [4](a))^-1]^-2:(x-a)/(4sqrt (x)+4sqrt (a))

We need to simplify the given expression step by step. The expression is:<br /><br />\[<br />\left[ (\sqrt[4]{x} - \sqrt[4]{a})^{-1} + (\sqrt[4]{x} + \sqrt[4]{a})^{-1} \right]^{-2} : \frac{x-a}{4\sqrt{x} + 4\sqrt{a}}<br />\]<br /><br />---<br /><br />### Step 1: Simplify the first part inside the brackets<br />The term inside the brackets is:<br /><br />\[<br />(\sqrt[4]{x} - \sqrt[4]{a})^{-1} + (\sqrt[4]{x} + \sqrt[4]{a})^{-1}<br />\]<br /><br />Using the property of reciprocals, we rewrite this as:<br /><br />\[<br />\frac{1}{\sqrt[4]{x} - \sqrt[4]{a}} + \frac{1}{\sqrt[4]{x} + \sqrt[4]{a}}<br />\]<br /><br />Take the least common denominator (LCD), which is \((\sqrt[4]{x} - \sqrt[4]{a})(\sqrt[4]{x} + \sqrt[4]{a})\). Using the difference of squares formula, the denominator becomes:<br /><br />\[<br />(\sqrt[4]{x})^2 - (\sqrt[4]{a})^2 = \sqrt{x} - \sqrt{a}<br />\]<br /><br />Thus, the sum becomes:<br /><br />\[<br />\frac{\sqrt[4]{x} + \sqrt[4]{a} + \sqrt[4]{x} - \sqrt[4]{a}}{\sqrt{x} - \sqrt{a}}<br />\]<br /><br />Simplify the numerator:<br /><br />\[<br />\sqrt[4]{x} + \sqrt[4]{a} + \sqrt[4]{x} - \sqrt[4]{a} = 2\sqrt[4]{x}<br />\]<br /><br />So the expression simplifies to:<br /><br />\[<br />\frac{2\sqrt[4]{x}}{\sqrt{x} - \sqrt{a}}<br />\]<br /><br />---<br /><br />### Step 2: Raise the result to the power of \(-2\)<br />Now raise the above result to the power of \(-2\):<br /><br />\[<br />\left( \frac{2\sqrt[4]{x}}{\sqrt{x} - \sqrt{a}} \right)^{-2}<br />\]<br /><br />Using the property of exponents, \((\frac{a}{b})^{-n} = (\frac{b}{a})^n\), we get:<br /><br />\[<br />\left( \frac{\sqrt{x} - \sqrt{a}}{2\sqrt[4]{x}} \right)^2<br />\]<br /><br />Square both the numerator and denominator:<br /><br />\[<br />\frac{(\sqrt{x} - \sqrt{a})^2}{(2\sqrt[4]{x})^2}<br />\]<br /><br />Simplify the denominator:<br /><br />\[<br />(2\sqrt[4]{x})^2 = 4\sqrt[4]{x}^2 = 4\sqrt{x}<br />\]<br /><br />So the result becomes:<br /><br />\[<br />\frac{(\sqrt{x} - \sqrt{a})^2}{4\sqrt{x}}<br />\]<br /><br />---<br /><br />### Step 3: Divide by the second term<br />Now divide the above result by the second term in the original expression:<br /><br />\[<br />\frac{x-a}{4\sqrt{x} + 4\sqrt{a}}<br />\]<br /><br />Dividing by a fraction is equivalent to multiplying by its reciprocal. So we multiply:<br /><br />\[<br />\frac{(\sqrt{x} - \sqrt{a})^2}{4\sqrt{x}} \times \frac{4\sqrt{x} + 4\sqrt{a}}{x-a}<br />\]<br /><br />Factorize \(x-a\) in the denominator using the difference of squares:<br /><br />\[<br />x-a = (\sqrt{x} - \sqrt{a})(\sqrt{x} + \sqrt{a})<br />\]<br /><br />Substitute this into the expression:<br /><br />\[<br />\frac{(\sqrt{x} - \sqrt{a})^2}{4\sqrt{x}} \times \frac{4\sqrt{x} + 4\sqrt{a}}{(\sqrt{x} - \sqrt{a})(\sqrt{x} + \sqrt{a})}<br />\]<br /><br />Cancel one \((\sqrt{x} - \sqrt{a})\) from the numerator and denominator:<br /><br />\[<br />\frac{\sqrt{x} - \sqrt{a}}{4\sqrt{x}} \times \frac{4\sqrt{x} + 4\sqrt{a}}{\sqrt{x} + \sqrt{a}}<br />\]<br /><br />Simplify further. Cancel \(4\sqrt{x}\) in the numerator and denominator:<br /><br />\[<br />\frac{\sqrt{x} - \sqrt{a}}{\sqrt{x} + \sqrt{a}} \times (\sqrt{x} + \sqrt{a})<br />\]<br /><br />Cancel \((\sqrt{x} + \sqrt{a})\):<br /><br />\[<br />\sqrt{x} - \sqrt{a}<br />\]<br /><br />---<br /><br />### Final Answer:<br />\[<br />\boxed{\sqrt{x} - \sqrt{a}}<br />\]