what is the following simplified product? assume $xgeq0$.\n$(sqrt{10x^{4}} - xsqrt{5x^{2}})(2sqrt{15x^{4}}+sq…

Explanation:

Step1: Expand using FOIL method

[ \begin{align*} &(\sqrt{10x^{4}}-x\sqrt{5x^{2}})(2\sqrt{15x^{4}}+\sqrt{3x^{3}})\ =&\sqrt{10x^{4}}\times2\sqrt{15x^{4}}+\sqrt{10x^{4}}\times\sqrt{3x^{3}}-x\sqrt{5x^{2}}\times2\sqrt{15x^{4}}-x\sqrt{5x^{2}}\times\sqrt{3x^{3}} \end{align*} ]

Step2: Simplify each term

Term 1:

[ \begin{align*} \sqrt{10x^{4}}\times2\sqrt{15x^{4}}&=2\sqrt{10x^{4}\times15x^{4}}\ &=2\sqrt{150x^{8}}\ &=2\sqrt{25\times6x^{8}}\ &=2\times5x^{4}\sqrt{6}\ & = 10x^{4}\sqrt{6} \end{align*} ]

Term 2:

[ \begin{align*} \sqrt{10x^{4}}\times\sqrt{3x^{3}}&=\sqrt{10x^{4}\times3x^{3}}\ &=\sqrt{30x^{7}}\ &=x^{3}\sqrt{30x} \end{align*} ]

Term 3:

[ \begin{align*} -x\sqrt{5x^{2}}\times2\sqrt{15x^{4}}&=- 2x\sqrt{5x^{2}\times15x^{4}}\ &=-2x\sqrt{75x^{6}}\ &=-2x\sqrt{25\times3x^{6}}\ &=-2x\times5x^{3}\sqrt{3}\ &=-10x^{4}\sqrt{3} \end{align*} ]

Term 4:

[ \begin{align*} -x\sqrt{5x^{2}}\times\sqrt{3x^{3}}&=-x\sqrt{5x^{2}\times3x^{3}}\ &=-x\sqrt{15x^{5}}\ &=-x^{3}\sqrt{15x} \end{align*} ]

Step3: Combine terms

[10x^{4}\sqrt{6}+x^{3}\sqrt{30x}-10x^{4}\sqrt{3}-x^{3}\sqrt{15x}]

Answer:

(10x^{4}\sqrt{6}+x^{3}\sqrt{30x}-10x^{4}\sqrt{3}-x^{3}\sqrt{15x})