1. **State the problem:** Simplify the function $$f(x) = \frac{x^h - 1}{m} - \frac{\left(\sqrt{d \cdot t}\right)^h + d^h \left(x^h - t^h\right) - d \cdot t}{d}$$ 2. **Recall important rules:** - Powers and roots: $\left(\sqrt{a}\right)^h = a^{\frac{h}{2}}$ - Distribute powers over products: $(ab)^h = a^h b^h$ - Simplify complex fractions by combining terms over a common denominator. 3. **Rewrite the square root term:** $$\left(\sqrt{d \cdot t}\right)^h = (d \cdot t)^{\frac{h}{2}} = d^{\frac{h}{2}} t^{\frac{h}{2}}$$ 4. **Rewrite the numerator of the second fraction:** $$d^{\frac{h}{2}} t^{\frac{h}{2}} + d^h (x^h - t^h) - d t$$ 5. **Rewrite the entire function:** $$f(x) = \frac{x^h - 1}{m} - \frac{d^{\frac{h}{2}} t^{\frac{h}{2}} + d^h (x^h - t^h) - d t}{d}$$ 6. **Split the second fraction into separate terms:** $$f(x) = \frac{x^h - 1}{m} - \left(\frac{d^{\frac{h}{2}} t^{\frac{h}{2}}}{d} + \frac{d^h (x^h - t^h)}{d} - \frac{d t}{d}\right)$$ 7. **Simplify each term by dividing powers of $d$:** - $$\frac{d^{\frac{h}{2}} t^{\frac{h}{2}}}{d} = d^{\frac{h}{2} - 1} t^{\frac{h}{2}}$$ - $$\frac{d^h (x^h - t^h)}{d} = d^{h - 1} (x^h - t^h)$$ - $$\frac{d t}{d} = t$$ 8. **Rewrite $f(x)$ with simplified terms:** $$f(x) = \frac{x^h - 1}{m} - \left(d^{\frac{h}{2} - 1} t^{\frac{h}{2}} + d^{h - 1} (x^h - t^h) - t\right)$$ 9. **Distribute the minus sign:** $$f(x) = \frac{x^h - 1}{m} - d^{\frac{h}{2} - 1} t^{\frac{h}{2}} - d^{h - 1} (x^h - t^h) + t$$ 10. **Expand the term $d^{h - 1} (x^h - t^h)$:** $$f(x) = \frac{x^h - 1}{m} - d^{\frac{h}{2} - 1} t^{\frac{h}{2}} - d^{h - 1} x^h + d^{h - 1} t^h + t$$ This is the simplified form of the function $f(x)$. **Final answer:** $$f(x) = \frac{x^h - 1}{m} - d^{\frac{h}{2} - 1} t^{\frac{h}{2}} - d^{h - 1} x^h + d^{h - 1} t^h + t$$