Section 1.3 : Radicals
12. Simplify the following expression. Assume that \(x\), \(y\) and \(z\) are positive.
\[\sqrt[4]{{4{x^3}y}}\,\,\sqrt[4]{{8{x^2}{y^3}{z^5}}}\]Show All Steps Hide All Steps
Start SolutionRemember that when we have a product of two radicals with the same index in an expression we first need to combine them into one root before we start the simplification process.
\[\sqrt[4]{{4{x^3}y}}\,\,\sqrt[4]{{8{x^2}{y^3}{z^5}}} = \,\sqrt[4]{{\left( {4{x^3}y} \right)\left( {8{x^2}{y^3}{z^5}} \right)}} = \,\sqrt[4]{{32{x^5}{y^4}{z^5}}}\] Show Step 2Now that the expression has been written as a single radical we can proceed as we did in the earlier problems.
The radicand can be written as,
\[32{x^5}{y^4}{z^5} = \left( {{2^4}{x^4}{y^4}{z^4}} \right)\left( {2xz} \right)\] Show Step 3Now that we’ve gotten the radicand rewritten it’s easy to deal with the radical and get the expression in simplified radical form.
\[\sqrt[4]{{4{x^3}y}}\,\,\sqrt[4]{{8{x^2}{y^3}{z^5}}} = \,\sqrt[4]{{32{x^5}{y^4}{z^5}}} = \sqrt[4]{{{2^4}{x^4}{y^4}{z^4}}}\,\,\sqrt[4]{{2xz}} = \require{bbox} \bbox[2pt,border:1px solid black]{{2xyz\,\,\sqrt[4]{{2xz}}}}\]