1. **State the problem:** Simplify the expression $\left(\frac{5}{x}\right)^6 \left(\frac{1}{x}\right)^3$ using properties of exponents and write it in radical form. 2. **Recall the properties of exponents:** - When multiplying expressions with the same base, add the exponents: $a^m \cdot a^n = a^{m+n}$. - Power of a quotient: $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$. - Radical form: $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. 3. **Apply the power of a quotient:** $$\left(\frac{5}{x}\right)^6 = \frac{5^6}{x^6}$$ $$\left(\frac{1}{x}\right)^3 = \frac{1^3}{x^3} = \frac{1}{x^3}$$ 4. **Multiply the two expressions:** $$\frac{5^6}{x^6} \cdot \frac{1}{x^3} = \frac{5^6 \cdot 1}{x^6 \cdot x^3} = \frac{5^6}{x^{6+3}} = \frac{5^6}{x^9}$$ 5. **Simplify the numerator:** $$5^6 = 15625$$ 6. **Write the expression in exponential notation:** $$\frac{15625}{x^9}$$ 7. **Convert to radical form:** $$\frac{15625}{x^9} = \frac{15625}{\left(x^3\right)^3} = 15625 \cdot x^{-9} = 15625 \cdot \left(x^3\right)^{-3} = 15625 \cdot \frac{1}{\left(x^3\right)^3}$$ Alternatively, express $x^9$ as $\left(x^3\right)^3$ and write: $$\frac{15625}{x^9} = 15625 \cdot \sqrt[3]{\frac{1}{x^9}} = 15625 \cdot \sqrt[3]{x^{-9}} = 15625 \cdot \left(\sqrt[3]{x}\right)^{-9}$$ **Final simplified expression:** $$\frac{15625}{x^9}$$ **Radical form:** $$\frac{15625}{\left(\sqrt[3]{x}\right)^9}$$