Divide \( \frac{\sqrt{11^{5}}}{\sqrt[4]{11^{3}}} \) using fractional exponents

To simplify \( \frac{\sqrt{11^5}}{\sqrt[4]{11^3}} \) using fractional exponents, we start by expressing both the numerator and denominator in terms of exponent fractions. The square root can be expressed as an exponent of \( \frac{1}{2} \), so: \[ \sqrt{11^5} = (11^5)^{\frac{1}{2}} = 11^{\frac{5}{2}}. \] Next, the fourth root can be expressed as an exponent of \( \frac{1}{4} \), so: \[ \sqrt[4]{11^3} = (11^3)^{\frac{1}{4}} = 11^{\frac{3}{4}}. \] Now we can rewrite the original expression: \[ \frac{\sqrt{11^5}}{\sqrt[4]{11^3}} = \frac{11^{\frac{5}{2}}}{11^{\frac{3}{4}}}. \] According to the properties of exponents, when dividing powers with the same base, we subtract the exponents: \[ 11^{\frac{5}{2} - \frac{3}{4}}. \] To subtract the exponents, we need a common denominator. The least common denominator between 2 and 4 is 4, so we convert \( \frac{5}{2} \): \[ \frac{5}{2} = \frac{10}{4}. \] Now we can perform the subtraction: \[ \frac{10}{4} - \frac{3}{4} = \frac{7}{4}. \] Thus, we have: \[ 11^{\frac{7}{4}}. \] Ultimately, the simplified expression is: \[ \frac{\sqrt{11^{5}}}{\sqrt[4]{11^{3}}} = 11^{\frac{7}{4}}. \]