Solve the following exponential equations and inequalities step-by-step. 1. Solve $4^{3x+1} = 8^{x-1}$: - Express bases as powers of 2: $4 = 2^2$, $8 = 2^3$. - Rewrite equation: $(2^2)^{3x+1} = (2^3)^{x-1} \Rightarrow 2^{2(3x+1)} = 2^{3(x-1)}$. - Equate exponents: $2(3x+1) = 3(x-1)$. - Simplify: $6x + 2 = 3x - 3$. - Solve for $x$: $6x - 3x = -3 - 2 \Rightarrow 3x = -5 \Rightarrow x = -\frac{5}{3}$. 2. Solve $9^{3x} = 27^{x-2}$: - Express bases as powers of 3: $9 = 3^2$, $27 = 3^3$. - Rewrite equation: $(3^2)^{3x} = (3^3)^{x-2} \Rightarrow 3^{6x} = 3^{3x - 6}$. - Equate exponents: $6x = 3x -6$. - Solve for $x$: $6x - 3x = -6 \Rightarrow 3x = -6 \Rightarrow x = -2$. 3. Solve $8^{x-1} = 16^{3x}$: - Express bases as powers of 2: $8=2^3$, $16=2^4$. - Rewrite: $(2^3)^{x-1} = (2^4)^{3x} \Rightarrow 2^{3x - 3} = 2^{12x}$. - Equate exponents: $3x - 3 = 12x$. - Solve: $-3 = 9x \Rightarrow x = -\frac{1}{3}$. 4. Solve $9^{3x+1} = 27^{2x+1}$: - Bases as powers of 3: $9=3^2$, $27=3^3$. - Rewrite: $(3^2)^{3x+1} = (3^3)^{2x+1} \Rightarrow 3^{6x + 2} = 3^{6x + 3}$. - Equate exponents: $6x + 2 = 6x + 3$. - Simplify: $2 = 3$ no solution; inspect carefully: - Recalculate exponents: Left $= 2(3x +1) = 6x + 2$; Right $= 3(2x +1) = 6x + 3$. - Since $6x + 2 = 6x + 3$ has no solution, answer: no solution. 5. Solve $4^x = 0.0625$: - Express $0.0625$ as fraction: $\frac{1}{16} = 16^{-1}$. - Since $4 = 2^2$, $16 = 2^4$, rewrite: $4^x = 16^{-1} \Rightarrow (2^2)^x = (2^4)^{-1} \Rightarrow 2^{2x} = 2^{-4}$. - Equate exponents: $2x = -4$. - Solve: $x = -2$. 6. Solve $3^{-x} = \frac{1}{243}$: - Since $243 = 3^5$, we rewrite: $3^{-x} = 3^{-5}$. - Equate exponents: $-x = -5$. - Solve: $x = 5$. 7. Solve $8^{x+3} = \frac{1}{16}$: - $8 = 2^3$, $16=2^4$. - Rewrite: $(2^3)^{x+3} = 2^{-4} \Rightarrow 2^{3x + 9} = 2^{-4}$. - Equate exponents: $3x + 9 = -4$. - Solve: $3x = -13 \Rightarrow x = -\frac{13}{3}$. 8. Solve $\frac{1}{125} = 25^{x+5}$: - $125 = 5^3$, $25 = 5^2$. - Rewrite: $5^{-3} = (5^2)^{x+5} = 5^{2x + 10}$. - Equate exponents: $-3 = 2x + 10$. - Solve: $2x = -13 \Rightarrow x = -\frac{13}{2}$. 9. Solve $4^{3x+1} = 1$: - $4 = 2^2$, rewrite: $(2^2)^{3x +1} = 1 \Rightarrow 2^{6x + 2} = 1 = 2^0$. - Equate exponents: $6x + 2 = 0$. - Solve: $6x = -2 \Rightarrow x = -\frac{1}{3}$. 10. Solve $8^{x-1} = 2^{3x +8}$: - $8 = 2^3$, rewrite: $(2^3)^{x-1} = 2^{3x + 8} \Rightarrow 2^{3x -3} = 2^{3x +8}$. - Equate exponents: $3x - 3 = 3x + 8$. - Simplify: $-3 = 8$ no solution, contradiction implies no solution. 11. Solve inequality $(\frac{1}{3})^{-x} > 27^{2x+1}$: - Rewrite bases: $\frac{1}{3} = 3^{-1}$, $27 = 3^3$. - Expression: $(3^{-1})^{-x} > (3^3)^{2x+1} \Rightarrow 3^x > 3^{6x + 3}$. - Since base $3 > 1$, inequality direction maintained: $x > 6x + 3$. - Solve: $x - 6x > 3 \Rightarrow -5x > 3 \Rightarrow x < -\frac{3}{5}$. 12. Solve $(\frac{1}{4})^{3-2x} < 8^{-x}$: - $\frac{1}{4} = 4^{-1}$, $4 = 2^2$, $8=2^3$. - Rewrite: $(4^{-1})^{3-2x} < (2^3)^{-x} \Rightarrow 4^{-3 + 2x} < 2^{-3x}$. - Express $4 = 2^2$: $2^{2(-3 + 2x)} < 2^{-3x} \Rightarrow 2^{-6 + 4x} < 2^{-3x}$. - Base $2 > 1$ so exponent inequality direction preserved: $-6 + 4x < -3x$. - Solve: $4x + 3x < 6 \Rightarrow 7x < 6 \Rightarrow x < \frac{6}{7}$. 13. Solve $3^{\frac{x+3}{3}} > 81^{5+x}$: - $81 = 3^4$. - Rewrite: $3^{\frac{x+3}{3}} > (3^4)^{5+x} \Rightarrow 3^{\frac{x+3}{3}} > 3^{4(5+x)}$. - Base $3 > 1$ so: $\frac{x+3}{3} > 20 + 4x$. - Multiply both sides by 3: $x + 3 > 60 + 12x$. - Solve: $x - 12x > 60 - 3 \Rightarrow -11x > 57 \Rightarrow x < -\frac{57}{11}$. 14. Solve $(\frac{16}{25})^{x+x} < 1$: - Rewrite as $(\frac{16}{25})^{2x} < 1$. - Since $\frac{16}{25} < 1$, and positive, inequality $a^{k} < 1$ holds for $k > 0$. - So $2x > 0 \Rightarrow x > 0$. 15. Solve $9^{\sqrt{2x - 1}} \geq 27$: - $9 = 3^2$, $27 = 3^3$. - Rewrite: $(3^2)^{\sqrt{2x -1}} \geq 3^3 \Rightarrow 3^{2\sqrt{2x -1}} \geq 3^3$. - Base $3 > 1$ implies: $2\sqrt{2x -1} \geq 3$. - Divide by 2: $\sqrt{2x -1} \geq \frac{3}{2}$. - Square both sides (valid since $\sqrt{2x -1} \geq 0$): $2x -1 \geq \left(\frac{3}{2}\right)^2 = \frac{9}{4}$. - Solve: $2x \geq 1 + \frac{9}{4} = \frac{4}{4} + \frac{9}{4} = \frac{13}{4}$. $x \geq \frac{13}{8}$. 16. Solve $2^{k - 2} < 8$: - $8 = 2^3$. - Inequality: $2^{k - 2} < 2^3$. - Base $2 > 1$ implies: $k - 2 < 3$. - Solve: $k < 5$. 17. Solve $x^{-5/3} > \frac{1}{32}$: - $\frac{1}{32} = 2^{-5}$. - Rewrite: $x^{-5/3} > 2^{-5}$. - Assume $x > 0$ to apply log. - Take reciprocals exponent: $x^{-5/3} = (x^{1})^{-5/3}$. - Rewrite inequality: $x^{-5/3} > 2^{-5} \Rightarrow (x^{1})^{-5/3} > 2^{-5}$. - Taking logarithms or converting base could be complicated; instead rewrite: Let’s rewrite: $x^{-5/3} > 2^{-5} \Rightarrow \frac{1}{x^{5/3}} > \frac{1}{2^5}$. - Multiply both sides by positive $x^{5/3}$ (assuming $x > 0$): $1 > \frac{x^{5/3}}{2^5}$. - Multiply both sides by $2^5$: $2^5 > x^{5/3}$. - Take both sides to the power $\frac{3}{5}$ (positive): $2^{5 \times \frac{3}{5}} > x$. - Simplify exponent: $2^3 > x$, thus $8 > x$. - Domain $x > 0$, so solution is $0 < x < 8$. 18. Solve $(3x)^{-2/3} < 0.04$: - $0.04 = \frac{1}{25} = 25^{-1}$. - Rewrite: $(3x)^{-2/3} < 25^{-1}$. - Take reciprocal powers: $(3x)^{-2/3} = \frac{1}{(3x)^{2/3}} < \frac{1}{25}$. - Multiply both sides by $(3x)^{2/3}$ (assume $3x > 0$) $1 < \frac{(3x)^{2/3}}{25}$. - Multiply both sides by 25: $25 < (3x)^{2/3}$. - Raise both sides to the power $\frac{3}{2}$: $25^{3/2} < 3x$. - Compute $25^{3/2} = (\sqrt{25})^3 = 5^3 = 125$. - Solve: $125 < 3x \Rightarrow x > \frac{125}{3}$. 19. Solve $\frac{1}{2} > (\frac{x}{8})^{1/3}$: - Cube both sides (since $\frac{1}{2} > 0$): $\left(\frac{1}{2}\right)^3 > \frac{x}{8}$. - Compute left side: $\frac{1}{8} > \frac{x}{8}$. - Multiply both sides by 8: $1 > x$. - Also, cube root defined, so domain $x \geq 0$ (to keep principal root real for real numbers is not necessary for cube root; all real allowed). - Solution: $x < 1$. 20. Solve $x^{4/3} \leq 27$: - Raise both sides to power $\frac{3}{4}$: $x \leq 27^{3/4}$. - Compute $27^{3/4} = (3^3)^{3/4} = 3^{9/4} = 3^{2 + 1/4} = 3^2 \times 3^{1/4} = 9 \times 3^{1/4} \approx 9 \times 1.316 = 11.844$. - Since exponent is even power with cube root, solution domain $x \geq 0$. - So final solution $0 \leq x \leq 27^{3/4}$. Final solutions summary: 1. $x = -\frac{5}{3}$ 2. $x = -2$ 3. $x = -\frac{1}{3}$ 4. No solution 5. $x = -2$ 6. $x = 5$ 7. $x = -\frac{13}{3}$ 8. $x = -\frac{13}{2}$ 9. $x = -\frac{1}{3}$ 10. No solution 11. $x < -\frac{3}{5}$ 12. $x < \frac{6}{7}$ 13. $x < -\frac{57}{11}$ 14. $x > 0$ 15. $x \geq \frac{13}{8}$ 16. $k < 5$ 17. $0 < x < 8$ 18. $x > \frac{125}{3}$ 19. $x < 1$ 20. $0 \leq x \leq 27^{3/4}$