1. Problem: Find the value of $n$ in each of the following exponential equations. 2. Formula: For equations of the form $a^n = b$, rewrite $b$ as a power of $a$ if possible, then equate exponents: if $a^n = a^m$, then $n = m$. 3. Important rules: - Negative exponents mean reciprocals: $a^{-n} = \frac{1}{a^n}$. - Fractional exponents represent roots: $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. 4. Solve each: 8a. $(\frac{1}{3})^n = 81$. Rewrite $81 = 3^4$, so $(3^{-1})^n = 3^4$. Thus, $3^{-n} = 3^4 \Rightarrow -n = 4 \Rightarrow n = -4$. 8b. $(\frac{1}{2})^n = 8$. Rewrite $8 = 2^3$, so $(2^{-1})^n = 2^3$. Thus, $2^{-n} = 2^3 \Rightarrow -n = 3 \Rightarrow n = -3$. 8c. $(\frac{1}{2})^n = 4^{n+1}$. Rewrite $4 = 2^2$, so $2^{-n} = (2^2)^{n+1} = 2^{2(n+1)}$. Equate exponents: $-n = 2n + 2 \Rightarrow -n - 2n = 2 \Rightarrow -3n = 2 \Rightarrow n = -\frac{2}{3}$. 8d. $(\frac{1}{2})^n = 32$. Rewrite $32 = 2^5$, so $2^{-n} = 2^5$. Thus, $-n = 5 \Rightarrow n = -5$. 8e. $(\frac{1}{2})^{n+1} = 2$. Rewrite $2 = 2^1$, so $2^{-(n+1)} = 2^1$. Equate exponents: $-(n+1) = 1 \Rightarrow -n -1 = 1 \Rightarrow -n = 2 \Rightarrow n = -2$. 8f. $(\frac{1}{16})^n = 4$. Rewrite $\frac{1}{16} = 16^{-1} = (2^4)^{-1} = 2^{-4}$ and $4 = 2^2$. So, $(2^{-4})^n = 2^2 \Rightarrow 2^{-4n} = 2^2$. Equate exponents: $-4n = 2 \Rightarrow n = -\frac{1}{2}$. --- 9. Problem: Find the value of $x$ in each of the following exponential equations. 9a. $3^{-x} = 27$. Rewrite $27 = 3^3$, so $3^{-x} = 3^3$. Equate exponents: $-x = 3 \Rightarrow x = -3$. 9b. $4^{-x} = \frac{1}{16}$. Rewrite $16 = 4^2$, so $\frac{1}{16} = 4^{-2}$. Equate exponents: $-x = -2 \Rightarrow x = 2$. 9c. $2^{-x} = 128$. Rewrite $128 = 2^7$, so $2^{-x} = 2^7$. Equate exponents: $-x = 7 \Rightarrow x = -7$. 9d. $2^{x+3} = 64$. Rewrite $64 = 2^6$, so $2^{x+3} = 2^6$. Equate exponents: $x + 3 = 6 \Rightarrow x = 3$. 9e. $3^{x+1} = \frac{1}{81}$. Rewrite $81 = 3^4$, so $\frac{1}{81} = 3^{-4}$. Equate exponents: $x + 1 = -4 \Rightarrow x = -5$. 9f. $2^{-x} = \frac{1}{256}$. Rewrite $256 = 2^8$, so $\frac{1}{256} = 2^{-8}$. Equate exponents: $-x = -8 \Rightarrow x = 8$. Final answers: 8a. $n = -4$ 8b. $n = -3$ 8c. $n = -\frac{2}{3}$ 8d. $n = -5$ 8e. $n = -2$ 8f. $n = -\frac{1}{2}$ 9a. $x = -3$ 9b. $x = 2$ 9c. $x = -7$ 9d. $x = 3$ 9e. $x = -5$ 9f. $x = 8$