Answer:
[tex]\Huge\boxed{(\sqrt3)(\sqrt[5]4)=3^\frac{7}{10}=\sqrt[10]{3^7}=\sqrt[10]{2187}}[/tex]
Step-by-step explanation:
[tex]\text{Use}\ \sqrt[n]{a}=a^\frac{1}{n}\\--------------------\\\\\sqrt3=\sqrt[2]3=3^\frac{1}{2}\\\\\sqrt[5]3=3^\frac{1}{5}\\\\(\sqrt3)(\sqrt[5]3)=\left(3^\frac{1}{2}\right)\left(3^\frac{1}{5}\right)\\\\\scriptsize\text{use}\ a^n\cdot a^m=a^{n+m}\\\\=3^{\frac{1}{2}+\frac{1}{5}}\\------------\\\normalsize\text{We must find the common denominator.}\\\\\text{list of multiples of 2:}\ 0,\ 2,\ 4,\ 6,\ 8,\ \boxed{10},\ 12,\ ...\\\\\text{list of multiples of 5:}\ 0,\ 5,\ \boxed{10},\ 15,\ ...[/tex]
[tex]\text{therefore the common denominator of }\ \dfrac{1}{2}\ \text{and}\ \dfrac{1}{5}\ \text{is 10}.\\\\\dfrac{1}{2}=\dfrac{1\cdot5}{2\cdot5}=\dfrac{5}{10}\\\\\dfrac{1}{5}=\dfrac{1\cdot2}{5\cdot2}=\dfrac{2}{10}\\\\\dfrac{1}{2}+\dfrac{1}{5}=\dfrac{5}{10}+\dfrac{2}{10}=\dfrac{5+2}{10}=\dfrac{7}{10}\\\\\text{therefore}\\\\=3^{\frac{1}{2}+\frac{1}{5}}=3^\frac{7}{10}=\sqrt[10]{3^7}=\sqrt[10]{2187}[/tex]
