1. **State the problem:** Simplify the expression involving powers of $g$ given as $$\frac{g(x)}{g^9} = \frac{4}{g^9}$$ and related expressions using exponent rules. 2. **Recall the exponent rules:** - Power of a Product: $ (ab)^m = a^m b^m $ - Power of a Power: $ (a^m)^n = a^{mn} $ - Quotient of Powers: $ \frac{a^m}{a^n} = a^{m-n} $ - Product of Powers: $ a^m \cdot a^n = a^{m+n} $ 3. **Analyze the given expression:** Given $$\frac{g(x)}{g^9} = \frac{4}{g^9}$$ implies $g(x) = 4$ since the denominators are equal. 4. **Simplify the expression $\frac{1}{g^9} = g$:** Rewrite as $$g \cdot g^9 = 1$$ Using Product of Powers: $$g^{1+9} = g^{10} = 1$$ 5. **Solve for $g$:** Since $g^{10} = 1$, $g$ is a 10th root of unity. The simplest real solution is: $$g = 1$$ 6. **Check the expression $7x \cdot \frac{1}{g^9} = \frac{4}{g^9}$:** Substitute $\frac{1}{g^9} = g$: $$7x \cdot g = 4g$$ Divide both sides by $g$ (assuming $g \neq 0$): $$\cancel{g} \cdot 7x = 4 \cancel{g}$$ $$7x = 4$$ 7. **Solve for $x$:** $$x = \frac{4}{7}$$ **Final answers:** $$g = 1$$ $$x = \frac{4}{7}$$