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Section 1.3 : Radicals

For problems 1 – 6 write the expression in exponential form.

1. \(\sqrt {3n} \)
2. \(\sqrt[6]{{2y}}\)
3. \(\sqrt[5]{{7{x^3}}}\)
4. \(\sqrt[4]{{xyz}}\)
5. \(\sqrt {x + y} \)
6. \(\sqrt[3]{{{a^3} + {b^3}}}\)

For problems 7 – 12 evaluate the radical.

1. \(\sqrt {256} \)
2. \(\sqrt[4]{{256}}\)
3. \(\sqrt[8]{{256}}\)
4. \(\sqrt[5]{{ - 1024}}\)
5. \(\sqrt[3]{{ - 216}}\)
6. \(\sqrt[3]{{343}}\)

For problems 13 – 22 simplify each of the following. Assume that \(x\), \(y\) and \(z\) are all positive.

1. \(\sqrt {{z^5}} \)
2. \(\sqrt[3]{{{z^5}}}\)
3. \(\sqrt[3]{{16{x^{17}}}}\)
4. \(\sqrt[6]{{128{y^{11}}}}\)
5. \(\sqrt {{x^3}{y^{17}}{z^4}} \)
6. \(\sqrt[4]{{{x^3}{y^{20}}{z^5}}}\)
7. \(\sqrt[4]{{729{x^7}y\,{z^{13}}}}\)
8. \(\sqrt[3]{{4{x^2}y}}\,\,\,\sqrt[3]{{10{x^5}{y^2}}}\)
9. \(\sqrt {3x} \,\,\sqrt {6x} \,\,\sqrt {14x} \)
10. \(\sqrt[4]{{2x{y^3}}}\,\,\,\sqrt[4]{{32{x^2}{y^2}}}\)

For problems 23 – 26 multiply each of the following. Assume that \(x\) is positive.

1. \(\left( {2\sqrt x + 4} \right)\left( {\sqrt x - 7} \right)\)
2. \(\sqrt[3]{x}\left( {\sqrt[3]{x} + 2\sqrt[3]{{{x^4}}}} \right)\)
3. \(\left( {\sqrt x + \sqrt {2y} } \right)\left( {\sqrt x - \sqrt {2y} } \right)\)
4. \({\left( {\sqrt[4]{x} + \sqrt[4]{{{x^2}}}} \right)^2}\)

For problems 27 – 35 rationalize the denominator. Assume that \(x\) and \(y\) are both positive.

1. \(\displaystyle \frac{9}{{\sqrt y }}\)
2. \(\displaystyle \frac{3}{{\sqrt {7x} }}\)
3. \(\displaystyle \frac{1}{{\sqrt[4]{x}}}\)
4. \(\displaystyle \frac{{12}}{{\sqrt[5]{{3{x^2}}}}}\)
5. \(\displaystyle \frac{2}{{4 - \sqrt x }}\)
6. \(\displaystyle \frac{9}{{\sqrt {3y} + 2}}\)
7. \(\displaystyle \frac{4}{{\sqrt 7 - 6\sqrt x }}\)
8. \(\displaystyle \frac{{ - 6}}{{\sqrt {5x} + 10\sqrt y }}\)
9. \(\displaystyle \frac{{4 + x}}{{x - \sqrt x }}\)

For problems 36 – 38 determine if the statement is true or false. If it is false explain why it is false.

1. \(3{x^{\frac{1}{2}}} = \sqrt {3x} \)
2. \(\sqrt[3]{{x + 6}} = \sqrt[3]{x} + \sqrt[3]{6}\)
3. \(\sqrt[4]{{{x^2}}} = \sqrt x \)
4. For problems 13 – 35 above we always added the instruction to assume that the variables were positive. Why was this instruction added? How would the answers to the problems change if we did not have that instruction?