I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 1.3 : Radicals
For problems 1 – 6 write the expression in exponential form.
- \(\sqrt {3n} \)
- \(\sqrt[6]{{2y}}\)
- \(\sqrt[5]{{7{x^3}}}\)
- \(\sqrt[4]{{xyz}}\)
- \(\sqrt {x + y} \)
- \(\sqrt[3]{{{a^3} + {b^3}}}\)
For problems 7 – 12 evaluate the radical.
- \(\sqrt {256} \)
- \(\sqrt[4]{{256}}\)
- \(\sqrt[8]{{256}}\)
- \(\sqrt[5]{{ - 1024}}\)
- \(\sqrt[3]{{ - 216}}\)
- \(\sqrt[3]{{343}}\)
For problems 13 – 22 simplify each of the following. Assume that \(x\), \(y\) and \(z\) are all positive.
- \(\sqrt {{z^5}} \)
- \(\sqrt[3]{{{z^5}}}\)
- \(\sqrt[3]{{16{x^{17}}}}\)
- \(\sqrt[6]{{128{y^{11}}}}\)
- \(\sqrt {{x^3}{y^{17}}{z^4}} \)
- \(\sqrt[4]{{{x^3}{y^{20}}{z^5}}}\)
- \(\sqrt[4]{{729{x^7}y\,{z^{13}}}}\)
- \(\sqrt[3]{{4{x^2}y}}\,\,\,\sqrt[3]{{10{x^5}{y^2}}}\)
- \(\sqrt {3x} \,\,\sqrt {6x} \,\,\sqrt {14x} \)
- \(\sqrt[4]{{2x{y^3}}}\,\,\,\sqrt[4]{{32{x^2}{y^2}}}\)
For problems 23 – 26 multiply each of the following. Assume that \(x\) is positive.
- \(\left( {2\sqrt x + 4} \right)\left( {\sqrt x - 7} \right)\)
- \(\sqrt[3]{x}\left( {\sqrt[3]{x} + 2\sqrt[3]{{{x^4}}}} \right)\)
- \(\left( {\sqrt x + \sqrt {2y} } \right)\left( {\sqrt x - \sqrt {2y} } \right)\)
- \({\left( {\sqrt[4]{x} + \sqrt[4]{{{x^2}}}} \right)^2}\)
For problems 27 – 35 rationalize the denominator. Assume that \(x\) and \(y\) are both positive.
- \(\displaystyle \frac{9}{{\sqrt y }}\)
- \(\displaystyle \frac{3}{{\sqrt {7x} }}\)
- \(\displaystyle \frac{1}{{\sqrt[4]{x}}}\)
- \(\displaystyle \frac{{12}}{{\sqrt[5]{{3{x^2}}}}}\)
- \(\displaystyle \frac{2}{{4 - \sqrt x }}\)
- \(\displaystyle \frac{9}{{\sqrt {3y} + 2}}\)
- \(\displaystyle \frac{4}{{\sqrt 7 - 6\sqrt x }}\)
- \(\displaystyle \frac{{ - 6}}{{\sqrt {5x} + 10\sqrt y }}\)
- \(\displaystyle \frac{{4 + x}}{{x - \sqrt x }}\)
For problems 36 – 38 determine if the statement is true or false. If it is false explain why it is false.
- \(3{x^{\frac{1}{2}}} = \sqrt {3x} \)
- \(\sqrt[3]{{x + 6}} = \sqrt[3]{x} + \sqrt[3]{6}\)
- \(\sqrt[4]{{{x^2}}} = \sqrt x \)
- 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?