Answer:
[tex]\boxed{\bold{3x^2y\sqrt[3]{y}}}[/tex]
Explanation:
[tex]\bold{\sqrt[3]{27x^6y^4}}[/tex]
[tex]\bold{\mathrm{Apply\:radical\:rule\:}\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b},\:\quad \mathrm{\:assuming\:}a\ge 0,\:b\ge 0}[/tex]
[tex]\bold{=\sqrt[3]{27}\sqrt[3]{x^6}\sqrt[3]{y^4}}[/tex]
[tex]\bold{\sqrt[3]{27} \ = \ 3}[/tex]
[tex]\bol=3\sqrt[3]{x^6}\sqrt[3]{y^4}d[/tex]
[tex]\bol\mathrm{Apply\:exponent\:rule}\quad \:a^{bc}=\left(a^b\right)^cd: \ x^6=x^{2\cdot \:3}=\left(x^2\right)^3[/tex]
[tex]\bol=3\sqrt[3]{\left(x^2\right)^3}\sqrt[3]{y^4}d[/tex]
[tex]\bold{\mathrm{Apply\:radical\:rule\:}\sqrt[n]{a^n}=a,\:\quad \mathrm{\:assuming\:}a\ge 0: \ \sqrt[3]{\left(x^2\right)^3}=x^2}[/tex]
[tex]\bold{=3x^2\sqrt[3]{y^4}}[/tex]
[tex]\bold{\mathrm{Apply\:exponent\:rule}:\quad \:a^{b+c}=a^ba^c}[/tex]
[tex]\bold{=3x^2\sqrt[3]{y^3y^1}}[/tex]
[tex]\bold{\mathrm{Apply\:radical\:rule}:\quad \sqrt{a\cdot b}=\sqrt{a}\sqrt{b}}[/tex]
[tex]\bold{=3x^2\sqrt[3]{y^3}\sqrt[3]{y^1}}[/tex]
[tex]\bold{\mathrm{Apply\:radical\:rule\:}\sqrt[n]{a^n}=a,\:\quad \mathrm{\:assuming\:}a\ge 0: \ \sqrt[3]{y^3}=y}[/tex]
[tex]\bold{=3x^2y\sqrt[3]{y^1}}[/tex]
[tex]\bold{\mathrm{Apply\:rule}\:a^1=a: \ y^1=y}[/tex]
[tex]\boxed{\bold{3x^2y\sqrt[3]{y}}}[/tex]
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