what is the simplest form of $\frac{x^{4}y^{7}}{sqrt3{x^{10}y^{4}}}$?\n$xy^{6}$\n$x^{9}y^{9}$\n$x^{8}y^{9}sqr…

Explanation:

Step1: Rewrite the denominator

Use the rule $\sqrt[n]{a^m}=a^{\frac{m}{n}}$. So, $\sqrt[3]{x^{10}y^{4}}=x^{\frac{10}{3}}y^{\frac{4}{3}}$.

Step2: Rewrite the fraction

The original expression $\frac{x^{4}y^{7}}{\sqrt[3]{x^{10}y^{4}}}$ becomes $\frac{x^{4}y^{7}}{x^{\frac{10}{3}}y^{\frac{4}{3}}}$.

Step3: Use the division - rule of exponents

The division - rule of exponents is $\frac{a^m}{a^n}=a^{m - n}$. For the $x$ terms: $x^{4-\frac{10}{3}}=x^{\frac{12 - 10}{3}}=x^{\frac{2}{3}}$. For the $y$ terms: $y^{7-\frac{4}{3}}=y^{\frac{21 - 4}{3}}=y^{\frac{17}{3}}$.

Step4: Rewrite in radical form

$x^{\frac{2}{3}}y^{\frac{17}{3}}=y^{5}\sqrt[3]{x^{2}y^{2}}$ since $y^{\frac{17}{3}}=y^{5+\frac{2}{3}}=y^{5}y^{\frac{2}{3}}$.

Answer:

$y^{5}\sqrt[3]{x^{2}y^{2}}$