1. **Problem:** Find $n$ if $(300)(400) = 12 \times 10^n$. 2. **Formula:** Express numbers in powers of 10 to compare. 3. **Work:** $(300)(400) = 300 \times 400 = 120000 = 12 \times 10000 = 12 \times 10^4$ 4. **Conclusion:** $n = 4$. --- 1. **Problem:** Find $x$ if $x^6 = 4^6$. 2. **Formula:** If $a^m = b^m$, then $a = b$ or $a = -b$ if $m$ is even. 3. **Work:** $x^6 = 4^6 \implies x = \pm 4$ 4. **Conclusion:** $x = 4$ or $x = -4$. --- 1. **Problem:** Simplify $\frac{27^{n+2} - 6 \times 3^{3n+3}}{3^n \times 9^{n+2}}$. 2. **Formula:** Express all terms with base 3. 3. **Work:** $27 = 3^3$, $9 = 3^2$ Numerator: $27^{n+2} = (3^3)^{n+2} = 3^{3n+6}$ $6 \times 3^{3n+3}$ stays as is. Denominator: $3^n \times 9^{n+2} = 3^n \times (3^2)^{n+2} = 3^n \times 3^{2n+4} = 3^{3n+4}$ Expression: $\frac{3^{3n+6} - 6 \times 3^{3n+3}}{3^{3n+4}} = 3^{(3n+6)-(3n+4)} - 6 \times 3^{(3n+3)-(3n+4)} = 3^2 - 6 \times 3^{-1} = 9 - 6 \times \frac{1}{3} = 9 - 2 = 7$ 4. **Conclusion:** The expression simplifies to $7$. --- 1. **Problem:** Solve for $m$ and $p$ given $(9^m)(5^{-2p}) = \frac{1}{15}$. 2. **Formula:** Express $\frac{1}{15}$ as $15^{-1} = (3 \times 5)^{-1} = 3^{-1} \times 5^{-1}$. 3. **Work:** $9^m = (3^2)^m = 3^{2m}$ Equation: $3^{2m} \times 5^{-2p} = 3^{-1} \times 5^{-1}$ Equate powers of bases: $3: 2m = -1 \implies m = -\frac{1}{2}$ $5: -2p = -1 \implies p = \frac{1}{2}$ 4. **Conclusion:** $m = -\frac{1}{2}$, $p = \frac{1}{2}$. --- 1. **Problem:** Find $\frac{x}{y}$ if $3^x \times 5^{-y} = 225$. 2. **Formula:** Express 225 as powers of 3 and 5. 3. **Work:** $225 = 15^2 = (3 \times 5)^2 = 3^2 \times 5^2$ Equation: $3^x \times 5^{-y} = 3^2 \times 5^2$ Equate powers: $x = 2$ $-y = 2 \implies y = -2$ Calculate $\frac{x}{y} = \frac{2}{-2} = -1$ 4. **Conclusion:** $\frac{x}{y} = -1$. --- 1. **Problem:** Solve for $x$ if $(4^{x+3})(16^x) = 8^{3x}$. 2. **Formula:** Express all bases as powers of 2. 3. **Work:** $4 = 2^2$, $16 = 2^4$, $8 = 2^3$ Rewrite: $(2^2)^{x+3} \times (2^4)^x = (2^3)^{3x}$ Simplify exponents: $2^{2x+6} \times 2^{4x} = 2^{9x}$ Combine left side: $2^{2x+6+4x} = 2^{6x+6}$ Equation: $2^{6x+6} = 2^{9x}$ Equate exponents: $6x + 6 = 9x \implies 6 = 3x \implies x = 2$ 4. **Conclusion:** $x = 2$. --- 1. **Problem:** Solve for $x$ if $\sqrt{3^{x+2}} + 17 = 8$. 2. **Formula:** $\sqrt{a} = a^{1/2}$. 3. **Work:** $\sqrt{3^{x+2}} = (3^{x+2})^{1/2} = 3^{\frac{x+2}{2}}$ Equation: $3^{\frac{x+2}{2}} + 17 = 8 \implies 3^{\frac{x+2}{2}} = 8 - 17 = -9$ Since $3^{\text{any real}} > 0$, no real solution. 4. **Conclusion:** No real solution for $x$. --- 1. **Problem:** Simplify $\frac{2^{18} - 2^{15} + 7}{2^{15} + 1}$. 2. **Formula:** Factor powers of 2. 3. **Work:** Factor numerator: $2^{15}(2^3 - 1) + 7 = 2^{15} \times 7 + 7 = 7(2^{15} + 1)$ Expression: $\frac{7(2^{15} + 1)}{2^{15} + 1} = 7$ 4. **Conclusion:** Simplifies to $7$. --- 1. **Problem:** Given $(2^{x-1})(3^{y+1}) = (3^4)(2^5)$, find (i) $x + y$ and (ii) $\frac{y}{x}$. 2. **Formula:** Equate powers of same bases. 3. **Work:** $2^{x-1} = 2^5 \implies x - 1 = 5 \implies x = 6$ $3^{y+1} = 3^4 \implies y + 1 = 4 \implies y = 3$ (i) $x + y = 6 + 3 = 9$ (ii) $\frac{y}{x} = \frac{3}{6} = \frac{1}{2}$ 4. **Conclusion:** (i) $9$, (ii) $\frac{1}{2}$. --- 1. **Problem:** Solve for $y$ if $\left(\frac{1}{9}\right)^{2y} \times \left(\frac{1}{3}\right)^y \div \frac{1}{27} = 3^{-5y}$. 2. **Formula:** Express all terms as powers of 3. 3. **Work:** $\frac{1}{9} = 3^{-2}$, $\frac{1}{3} = 3^{-1}$, $\frac{1}{27} = 3^{-3}$ Rewrite: $(3^{-2})^{2y} \times (3^{-1})^y \div 3^{-3} = 3^{-5y}$ Simplify exponents: $3^{-4y} \times 3^{-y} \times 3^{3} = 3^{-5y}$ Combine left side: $3^{-4y - y + 3} = 3^{-5y}$ Simplify exponent: $3^{-5y + 3} = 3^{-5y}$ Equate exponents: $-5y + 3 = -5y \implies 3 = 0$ Contradiction means no solution unless re-check. Check division carefully: Dividing by $\frac{1}{27}$ is multiplying by $27 = 3^3$. So correct: $3^{-4y} \times 3^{-y} \times 3^{3} = 3^{-5y}$ Left side exponent: $-4y - y + 3 = -5y + 3$ Equation: $3^{-5y + 3} = 3^{-5y}$ Equate exponents: $-5y + 3 = -5y \implies 3 = 0$ no solution. This means no real $y$ satisfies the equation. 4. **Conclusion:** No solution for $y$.