1. **Problem:** Simplify the expression $$\frac{(mm^2)^5 - 2}{(3 \times 5^2)} \times \frac{(atb)^3 (t+d)^3}{(atb)(t+d)^3}$$ 2. **Step 1: Simplify each part separately.** - Simplify $$(mm^2)^5$$: Since $m$ is a variable, $mm^2 = m^{1+2} = m^3$, so $$(mm^2)^5 = (m^3)^5 = m^{3 \times 5} = m^{15}$$. - So numerator becomes $$m^{15} - 2$$. - Simplify denominator of first fraction: $$3 \times 5^2 = 3 \times 25 = 75$$. 3. **Step 2: Simplify the second fraction.** - Numerator: $$(atb)^3 (t+d)^3$$. - Denominator: $$(atb)(t+d)^3$$. - Cancel common terms: $$(atb)^3 / (atb) = (atb)^{3-1} = (atb)^2$$. - $$(t+d)^3 / (t+d)^3 = 1$$. - So the second fraction simplifies to $$(atb)^2$$. 4. **Step 3: Combine all parts.** $$\frac{m^{15} - 2}{75} \times (atb)^2 = \frac{(m^{15} - 2)(atb)^2}{75}$$. **Final simplified expression:** $$\boxed{\frac{(m^{15} - 2)(atb)^2}{75}}$$