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12. Simplify the following expression. Assume that $$x$$, $$y$$ and $$z$$ are positive.

$\sqrt{{4{x^3}y}}\,\,\sqrt{{8{x^2}{y^3}{z^5}}}$

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Remember 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.

$\sqrt{{4{x^3}y}}\,\,\sqrt{{8{x^2}{y^3}{z^5}}} = \,\sqrt{{\left( {4{x^3}y} \right)\left( {8{x^2}{y^3}{z^5}} \right)}} = \,\sqrt{{32{x^5}{y^4}{z^5}}}$ Show Step 2

Now 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 3

Now that we’ve gotten the radicand rewritten it’s easy to deal with the radical and get the expression in simplified radical form.

$\sqrt{{4{x^3}y}}\,\,\sqrt{{8{x^2}{y^3}{z^5}}} = \,\sqrt{{32{x^5}{y^4}{z^5}}} = \sqrt{{{2^4}{x^4}{y^4}{z^4}}}\,\,\sqrt{{2xz}} = \require{bbox} \bbox[2pt,border:1px solid black]{{2xyz\,\,\sqrt{{2xz}}}}$