91Ó°ÊÓ

In Exercises \(5-52,\) express each of the given expressions in simplest form with only positive exponents. $$\frac{x-y^{-1}}{x^{-1}-y}$$

Simplify the given expressions. Express all answers with positive exponents. $$\frac{\left(x+x^{1 / 2}\right)\left(x-x^{1 / 2}\right)}{x}$$

Write each expression in simplest radical form. If \(a\) radical appears in the denominator, rationalize the denominator. \(\sqrt[5]{\sqrt[4]{9}}\)

Express each of the given expressions in simplest form with only positive exponents. $$\frac{x^{-2}-y^{-2}}{x^{-1}-y^{-1}}$$

Write each expression in simplest radical form. If \(a\) radical appears in the denominator, rationalize the denominator. \(\sqrt{\sqrt{\sqrt{n}}}\)

Perform the indicated operations, expressing answers in simplest form with rationalized denominators. Then verify the result with a calculator. $$(\sqrt{11}+\sqrt{6})(\sqrt{11}-2 \sqrt{6})$$

Solve the given problems. Find the exact sum of the positive roots of \(x^{2}-2 x-2=0\) and \(x^{2}+2 x-11=0\)

In Exercises \(5-52,\) express each of the given expressions in simplest form with only positive exponents. $$\frac{x^{-2}-y^{-2}}{x^{-1}-y^{-1}}$$

Simplify the given expressions. Express all answers with positive exponents. $$\frac{y^{3 / 8}\left(y^{5 / 8}-y^{13 / 8}\right)}{y^{1 / 2}\left(y^{1 / 2}-y^{-1 / 2}\right)}$$

Perform the indicated operations, expressing answers in simplest form with rationalized denominators. Then verify the result with a calculator. $$(2 \sqrt{5}-\sqrt{7})(3 \sqrt{5}+\sqrt{7})$$