1. **State the problem:** Simplify the expression $$\left( \frac{p^2 t^7}{10} \right)^3$$. 2. **Recall the exponent rule:** When raising a quotient to a power, apply the power to both numerator and denominator: $$\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}$$. 3. **Apply the rule:** $$\left( \frac{p^2 t^7}{10} \right)^3 = \frac{(p^2)^3 (t^7)^3}{10^3}$$. 4. **Use the power of a power rule:** $$ (x^m)^n = x^{m \times n} $$ So, $$ (p^2)^3 = p^{2 \times 3} = p^6 $$ $$ (t^7)^3 = t^{7 \times 3} = t^{21} $$ 5. **Calculate denominator:** $$ 10^3 = 10 \times 10 \times 10 = 1000 $$ 6. **Write the simplified expression:** $$ \frac{p^6 t^{21}}{1000} $$ **Final answer:** $$\boxed{\frac{p^6 t^{21}}{1000}}$$