1. **State the problem:** Evaluate the definite integral $$\int_7^8 x\sqrt{x} - 7 \, dx$$ with four decimal places. 2. **Rewrite the integrand:** Note that $$\sqrt{x} = x^{\frac{1}{2}}$$, so $$x\sqrt{x} = x \cdot x^{\frac{1}{2}} = x^{\frac{3}{2}}$$. 3. **Express the integral:** $$\int_7^8 x\sqrt{x} - 7 \, dx = \int_7^8 x^{\frac{3}{2}} - 7 \, dx = \int_7^8 x^{\frac{3}{2}} \, dx - \int_7^8 7 \, dx$$ 4. **Use the power rule for integration:** For $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$, where $$n \neq -1$$. 5. **Integrate each term:** $$\int x^{\frac{3}{2}} \, dx = \frac{x^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} = \frac{x^{\frac{5}{2}}}{\frac{5}{2}} = \frac{2}{5} x^{\frac{5}{2}}$$ $$\int 7 \, dx = 7x$$ 6. **Evaluate the definite integral:** $$\int_7^8 x^{\frac{3}{2}} - 7 \, dx = \left[ \frac{2}{5} x^{\frac{5}{2}} - 7x \right]_7^8 = \left( \frac{2}{5} 8^{\frac{5}{2}} - 7 \cdot 8 \right) - \left( \frac{2}{5} 7^{\frac{5}{2}} - 7 \cdot 7 \right)$$ 7. **Calculate powers:** $$8^{\frac{5}{2}} = (8^{\frac{1}{2}})^5 = (\sqrt{8})^5 = (2.8284)^5 \approx 181.0193$$ $$7^{\frac{5}{2}} = (7^{\frac{1}{2}})^5 = (\sqrt{7})^5 = (2.6458)^5 \approx 128.1292$$ 8. **Substitute values:** $$\frac{2}{5} \times 181.0193 - 7 \times 8 = 72.4077 - 56 = 16.4077$$ $$\frac{2}{5} \times 128.1292 - 7 \times 7 = 51.2517 - 49 = 2.2517$$ 9. **Final evaluation:** $$16.4077 - 2.2517 = 14.1560$$ **Answer:** $$\boxed{14.1560}$$