Answer:
[tex]4 x^{\frac{11}{10}} \cdot y^{\frac{17}{3}}[/tex]
Step-by-step explanation:
The given expression: [tex]4 \sqrt[5]{x^{3}} \cdot y^{4} \cdot \sqrt{x} \cdot \sqrt[3]{y^{5}}[/tex]
Step 1: Change radical to fractional exponent.
Formula for fractional exponent: [tex]\sqrt[n]{a}=a^{\frac{1}{n}}[/tex]
The power to which the base is raised becomes the numerator and the root becomes the denominator.
[tex]\Rightarrow 4 x^{\frac{3}{5}} \cdot y^{4} \cdot x^{\frac{1}{2}} \cdot y^{\frac{5}{3}}[/tex]
Step 2: Apply law of exponent for a product [tex]a^{m} \times a^{n}=a^{m+n}[/tex]
Multiply powers with same base.
[tex]\Rightarrow 4 x^{\frac{3}{5}+\frac{1}{2}} \cdot y^{4+\frac{5}{8}}[/tex]
Take LCM for the fractions in the power.
[tex]\Rightarrow 4 x^{\frac{6}{10}+\frac{5}{10}} \cdot y^{\frac{12}{3}+\frac{5}{3}}[/tex]
[tex]\Rightarrow 4 x^{\frac{11}{10}} \cdot y^{\frac{17}{3}}[/tex]
Hence the simplified form of [tex]4 \sqrt[5]{x^{3}} \cdot y^{4} \cdot \sqrt{x} \cdot \sqrt[3]{y^{5}} \text { is } 4 x^{\frac{11}{10}} \cdot y^{\frac{17}{3}}[/tex].
