Pauls Online Notes
Pauls Online Notes
Home / Algebra / Preliminaries / Radicals
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.
Assignment Problems Notice
Please do not email me to get solutions and/or answers to these problems. I will not give them out under any circumstances nor will I respond to any requests to do so. The intent of these problems is for instructors to use them for assignments and having solutions/answers easily available defeats that purpose.

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?