1. **Problem 37:** Simplify $$\left(\frac{-7a^2 b^3 c^0}{3a^3 b^4 c^3}\right)^{-4}$$. 2. **Recall the rules:** - Negative exponent rule: $$x^{-n} = \frac{1}{x^n}$$. - Power of a quotient: $$\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$$. - Power of a product: $$ (xy)^n = x^n y^n $$. - Zero exponent: $$c^0 = 1$$. 3. **Simplify inside the parentheses first:** $$\frac{-7a^2 b^3 c^0}{3a^3 b^4 c^3} = \frac{-7a^2 b^3 \cdot 1}{3a^3 b^4 c^3} = \frac{-7a^2 b^3}{3a^3 b^4 c^3}$$ 4. **Simplify variables by subtracting exponents:** - For $$a$$: $$a^{2-3} = a^{-1} = \frac{1}{a}$$ - For $$b$$: $$b^{3-4} = b^{-1} = \frac{1}{b}$$ - For $$c$$: $$c^{0-3} = c^{-3} = \frac{1}{c^3}$$ So the fraction becomes: $$\frac{-7}{3} \cdot a^{-1} \cdot b^{-1} \cdot c^{-3} = \frac{-7}{3 a b c^3}$$ 5. **Apply the negative exponent $$-4$$:** $$\left(\frac{-7}{3 a b c^3}\right)^{-4} = \left(\frac{3 a b c^3}{-7}\right)^4$$ 6. **Raise numerator and denominator to the 4th power:** $$\frac{(3)^4 (a)^4 (b)^4 (c^3)^4}{(-7)^4} = \frac{81 a^4 b^4 c^{12}}{2401}$$ 7. **Final answer for 37:** $$\boxed{\frac{81 a^4 b^4 c^{12}}{2401}}$$ --- 8. **Problem 38:** Simplify $$\left(\frac{-2a^3 b^2 c^0}{3a^2 b^3 c^7}\right)^{-2}$$. 9. **Simplify inside the parentheses:** $$\frac{-2a^3 b^2 \cdot 1}{3a^2 b^3 c^7} = \frac{-2a^3 b^2}{3a^2 b^3 c^7}$$ 10. **Simplify variables by subtracting exponents:** - For $$a$$: $$a^{3-2} = a^1 = a$$ - For $$b$$: $$b^{2-3} = b^{-1} = \frac{1}{b}$$ - For $$c$$: $$c^{0-7} = c^{-7} = \frac{1}{c^7}$$ So the fraction becomes: $$\frac{-2 a}{3 b c^7}$$ 11. **Apply the negative exponent $$-2$$:** $$\left(\frac{-2 a}{3 b c^7}\right)^{-2} = \left(\frac{3 b c^7}{-2 a}\right)^2$$ 12. **Raise numerator and denominator to the 2nd power:** $$\frac{3^2 b^2 c^{14}}{(-2)^2 a^2} = \frac{9 b^2 c^{14}}{4 a^2}$$ 13. **Final answer for 38:** $$\boxed{\frac{9 b^2 c^{14}}{4 a^2}}$$