1. **State the problem:** Solve the exponential equations in Set 2 by expressing both sides with common bases and then equate the exponents. 2. **Recall the rule:** If $a^m = a^n$ and $a > 0$, $a \neq 1$, then $m = n$. --- **RED:** $6^{3x-2} = 6^{2x}$ Set exponents equal: $$3x - 2 = 2x$$ Subtract $2x$: $$3x - 2x - 2 = 0 \Rightarrow x - 2 = 0$$ $$\Rightarrow x = 2$$ --- **ORANGE:** $1^{2x+9} \cdot 1^{7x+8} = 1^{8x+3}$ Since $1^k = 1$ for any $k$, both sides equal 1 for all $x$. Solution: All real $x$. --- **YELLOW:** $5^x \cdot 5^{x+3} = 5^{3x}$ Combine left side: $$5^{x + x + 3} = 5^{3x} \Rightarrow 5^{2x + 3} = 5^{3x}$$ Set exponents equal: $$2x + 3 = 3x$$ Subtract $2x$: $$3 = x$$ --- **LIGHT GREEN:** $3^{2x-7} = 27$ Express 27 as $3^3$: $$3^{2x - 7} = 3^3$$ Set exponents equal: $$2x - 7 = 3$$ Add 7: $$2x = 10$$ Divide by 2: $$x = 5$$ --- **DARK GREEN:** $240^{2x+1} = 343^{3x}$ 240 and 343 have no common base (240 factors to $2^4 \cdot 3 \cdot 5$, 343 is $7^3$). No solution by common base method. --- **LIGHT BLUE:** $(1/216)^{x-4} = (1/36)^x$ Rewrite bases: $$1/216 = 216^{-1} = (6^3)^{-1} = 6^{-3}$$ $$1/36 = 36^{-1} = (6^2)^{-1} = 6^{-2}$$ Rewrite equation: $$6^{-3(x-4)} = 6^{-2x}$$ Simplify exponents: $$6^{-3x + 12} = 6^{-2x}$$ Set exponents equal: $$-3x + 12 = -2x$$ Add $3x$: $$12 = x$$ --- **DARK BLUE:** $81^{3x+8} = 243^{2x+10}$ Rewrite bases: $$81 = 3^4$$ $$243 = 3^5$$ Rewrite equation: $$ (3^4)^{3x+8} = (3^5)^{2x+10}$$ $$3^{4(3x+8)} = 3^{5(2x+10)}$$ $$3^{12x + 32} = 3^{10x + 50}$$ Set exponents equal: $$12x + 32 = 10x + 50$$ Subtract $10x$: $$2x + 32 = 50$$ Subtract 32: $$2x = 18$$ Divide by 2: $$x = 9$$ --- **PURPLE:** $125^x \cdot 25^{-3x} = (1/125)^{2x+1}$ Rewrite bases: $$125 = 5^3$$ $$25 = 5^2$$ $$1/125 = 125^{-1} = 5^{-3}$$ Rewrite equation: $$5^{3x} \cdot 5^{2(-3x)} = 5^{-3(2x+1)}$$ $$5^{3x} \cdot 5^{-6x} = 5^{-6x - 3}$$ Combine left: $$5^{3x - 6x} = 5^{-6x - 3}$$ $$5^{-3x} = 5^{-6x - 3}$$ Set exponents equal: $$-3x = -6x - 3$$ Add $6x$: $$3x = -3$$ Divide by 3: $$x = -1$$ --- **PINK:** $32^{3x-1} \cdot (1/16)^x = 64$ Rewrite bases: $$32 = 2^5$$ $$16 = 2^4$$ $$64 = 2^6$$ Rewrite equation: $$2^{5(3x-1)} \cdot 2^{-4x} = 2^6$$ $$2^{15x - 5} \cdot 2^{-4x} = 2^6$$ Combine left: $$2^{15x - 5 - 4x} = 2^6$$ $$2^{11x - 5} = 2^6$$ Set exponents equal: $$11x - 5 = 6$$ Add 5: $$11x = 11$$ Divide by 11: $$x = 1$$ --- **BROWN:** $(1/729)^{x+4} \cdot 3^{x+2} = (1/9)^{x-1} \cdot 1/27$ Rewrite bases: $$729 = 3^6$$ $$9 = 3^2$$ $$27 = 3^3$$ Rewrite equation: $$3^{-6(x+4)} \cdot 3^{x+2} = 3^{-2(x-1)} \cdot 3^{-3}$$ Simplify exponents: $$3^{-6x - 24 + x + 2} = 3^{-2x + 2 - 3}$$ $$3^{-5x - 22} = 3^{-2x - 1}$$ Set exponents equal: $$-5x - 22 = -2x - 1$$ Add $5x$: $$-22 = 3x - 1$$ Add 1: $$-21 = 3x$$ Divide by 3: $$x = -7$$