1. **Problem:** Find the value of $t$ in the equation $3^{2t} = 9$. 2. **Formula and rules:** Recall that $9 = 3^2$, so we can write the equation as $3^{2t} = 3^2$. 3. **Step:** Since the bases are the same and nonzero, set the exponents equal: $$2t = 2$$ 4. **Solve for $t$:** $$t = \frac{2}{2} = 1$$ --- 1. **Problem:** Find the value of $x$ in the equation $x^3 \times x^4 \div x^2 = 2^5$. 2. **Formula and rules:** Use the laws of exponents: $a^m \times a^n = a^{m+n}$ and $\frac{a^m}{a^n} = a^{m-n}$. 3. **Step:** Simplify the left side: $$x^3 \times x^4 = x^{3+4} = x^7$$ 4. **Step:** Then divide by $x^2$: $$\frac{x^7}{x^2} = x^{7-2} = x^5$$ 5. **Equation becomes:** $$x^5 = 2^5$$ 6. **Since bases are equal, set:** $$x = 2$$ --- 1. **Problem:** Find the value of $y$ in the equation $5^{2y+1} = \frac{1}{25}$. 2. **Formula and rules:** Note that $25 = 5^2$, so $\frac{1}{25} = 5^{-2}$. 3. **Step:** Rewrite the equation: $$5^{2y+1} = 5^{-2}$$ 4. **Set exponents equal:** $$2y + 1 = -2$$ 5. **Solve for $y$:** $$2y = -2 - 1 = -3$$ $$y = \frac{-3}{2} = -1.5$$ **Final answers:** 1) $t = 1$ 2) $x = 2$ 3) $y = -1.5$