1. Simplify the following expressions involving indices. (i) Simplify $a^2 (2a^{-1})^2$: - Use the power of a product rule: $(xy)^n = x^n y^n$. - $(2a^{-1})^2 = 2^2 (a^{-1})^2 = 4a^{-2}$. - Multiply by $a^2$: $a^2 \times 4a^{-2} = 4a^{2 + (-2)} = 4a^0 = 4$. (ii) Simplify $2^n \times 8^{2n}$: - Express 8 as $2^3$: $8^{2n} = (2^3)^{2n} = 2^{6n}$. - Multiply powers with the same base: $2^n \times 2^{6n} = 2^{n + 6n} = 2^{7n}$. (iii) Simplify $5a^4 b^3 (a^2 b^5)^2$: - Apply power to each factor: $(a^2)^2 = a^4$, $(b^5)^2 = b^{10}$. - Expression becomes $5a^4 b^3 \times a^4 b^{10} = 5a^{4+4} b^{3+10} = 5a^8 b^{13}$. (iv) Simplify $\frac{x^4 (y^2 z^3)^4}{(x^2 z^{-5})^2}$: - Expand numerator: $(y^2)^4 = y^8$, $(z^3)^4 = z^{12}$, so numerator is $x^4 y^8 z^{12}$. - Expand denominator: $(x^2)^2 = x^4$, $(z^{-5})^2 = z^{-10}$, so denominator is $x^4 z^{-10}$. - Divide powers: $\frac{x^4 y^8 z^{12}}{x^4 z^{-10}} = y^8 z^{12 - (-10)} = y^8 z^{22}$. (v) Simplify $\frac{9^0 \times 81 \times 32^2}{3^8 \times 16^3}$: - $9^0 = 1$. - Express bases as powers of 3 and 2: $81 = 3^4$, $32 = 2^5$, $16 = 2^4$. - Numerator: $1 \times 3^4 \times (2^5)^2 = 3^4 \times 2^{10}$. - Denominator: $3^8 \times (2^4)^3 = 3^8 \times 2^{12}$. - Divide: $3^{4-8} \times 2^{10-12} = 3^{-4} \times 2^{-2} = \frac{1}{3^4} \times \frac{1}{2^2} = \frac{1}{81 \times 4} = \frac{1}{324}$. (vi) Simplify $\frac{8^2 \times 4^4}{32 \times 16^3}$: - Express bases as powers of 2: $8=2^3$, $4=2^2$, $32=2^5$, $16=2^4$. - Numerator: $(2^3)^2 \times (2^2)^4 = 2^6 \times 2^8 = 2^{14}$. - Denominator: $2^5 \times (2^4)^3 = 2^5 \times 2^{12} = 2^{17}$. - Divide: $2^{14-17} = 2^{-3} = \frac{1}{8}$. (vii) Simplify $\frac{15 x^{-2}}{x}$: - $x^{-2} / x = x^{-2 - 1} = x^{-3}$. - Expression: $15 x^{-3} = \frac{15}{x^3}$. (viii) Simplify $\frac{(9x)^{-3}}{3x^3}$: - $(9x)^{-3} = 9^{-3} x^{-3} = (3^2)^{-3} x^{-3} = 3^{-6} x^{-3}$. - Denominator: $3 x^3$. - Divide: $\frac{3^{-6} x^{-3}}{3 x^3} = 3^{-6 - 1} x^{-3 - 3} = 3^{-7} x^{-6} = \frac{1}{3^7 x^6}$. (ix) Simplify $\frac{2a^2 b^3}{(2ab)^{-2}}$: - Denominator: $(2ab)^{-2} = 2^{-2} a^{-2} b^{-2}$. - Expression: $2 a^2 b^3 \times 2^{2} a^{2} b^{2} = 2^{1+2} a^{2+2} b^{3+2} = 2^3 a^4 b^5 = 8 a^4 b^5$. (x) Simplify $\frac{25^{3n} \times 125}{625^n}$: - Express bases as powers of 5: $25=5^2$, $125=5^3$, $625=5^4$. - Numerator: $(5^2)^{3n} \times 5^3 = 5^{6n} \times 5^3 = 5^{6n + 3}$. - Denominator: $(5^4)^n = 5^{4n}$. - Divide: $5^{6n + 3 - 4n} = 5^{2n + 3}$. (xi) Simplify $\sqrt{\frac{2}{7} \times \frac{9}{4} a^4}$: - Multiply inside the root: $\frac{2}{7} \times \frac{9}{4} = \frac{18}{28} = \frac{9}{14}$. - So expression is $\sqrt{\frac{9}{14} a^4} = \sqrt{\frac{9}{14}} \times \sqrt{a^4} = \frac{3}{\sqrt{14}} a^2$. (xii) Simplify $(\frac{2}{5})^{-3} \times \frac{32}{25}$: - $(\frac{2}{5})^{-3} = (\frac{5}{2})^{3} = \frac{125}{8}$. - Multiply: $\frac{125}{8} \times \frac{32}{25} = \frac{125 \times 32}{8 \times 25} = \frac{4000}{200} = 20$. 2. Simplify expressions with fractional and negative indices. (i) $8^{1/3} = (2^3)^{1/3} = 2^{3 \times \frac{1}{3}} = 2^1 = 2$. (ii) $9^{3/2} = (3^2)^{3/2} = 3^{2 \times \frac{3}{2}} = 3^3 = 27$. (iii) $32^{-3/5} = (2^5)^{-3/5} = 2^{5 \times (-3/5)} = 2^{-3} = \frac{1}{8}$. (iv) $49^{-3/2} = (7^2)^{-3/2} = 7^{2 \times (-3/2)} = 7^{-3} = \frac{1}{343}$. (v) $(\sqrt[3]{5})^{10} = 5^{\frac{1}{3} \times 10} = 5^{10/3}$. (vi) $(\frac{16}{81})^{-1/4} = (\frac{2^4}{3^4})^{-1/4} = (\frac{2}{3})^{-1} = \frac{3}{2}$. (vii) Simplify $5^{n+1} \times 25^n \div 125^{(3n-1)/3}$: - Express bases as powers of 5: $25=5^2$, $125=5^3$. - Expression: $5^{n+1} \times (5^2)^n \div (5^3)^{(3n-1)/3} = 5^{n+1} \times 5^{2n} \div 5^{3n-1} = 5^{n+1+2n - (3n -1)} = 5^{n+1+2n - 3n +1} = 5^{2}$. (viii) Simplify $\frac{(32^{5/3} (2/3))^{-2}}{\sqrt{5/16}}$: - $32 = 2^5$, so $32^{5/3} = (2^5)^{5/3} = 2^{25/3}$. - Numerator inside parentheses: $2^{25/3} \times \frac{2}{3}$. - Raise to $-2$: $(2^{25/3} \times \frac{2}{3})^{-2} = 2^{-50/3} \times (\frac{2}{3})^{-2} = 2^{-50/3} \times (\frac{3}{2})^{2} = 2^{-50/3} \times \frac{9}{4}$. - Denominator: $\sqrt{\frac{5}{16}} = \frac{\sqrt{5}}{4}$. - Expression: $\frac{2^{-50/3} \times \frac{9}{4}}{\frac{\sqrt{5}}{4}} = 2^{-50/3} \times 9 \times \frac{1}{\sqrt{5}} = \frac{9}{\sqrt{5}} 2^{-50/3}$. (ix) Simplify $\frac{2^{2n+1} 3^{n+10}}{3^{n+1} 2^{n-1}}$: - Divide powers with same base: $2^{2n+1 - (n-1)} = 2^{n+2}$. - $3^{n+10 - (n+1)} = 3^{9}$. - Expression: $2^{n+2} \times 3^{9}$. (x) Simplify $\frac{3^{n+3}}{3^{n-1}} \div \frac{9^{n+1}}{3^{n-1}}$: - First fraction: $3^{n+3 - (n-1)} = 3^{4}$. - Second fraction: $\frac{9^{n+1}}{3^{n-1}} = \frac{(3^2)^{n+1}}{3^{n-1}} = \frac{3^{2n+2}}{3^{n-1}} = 3^{n+3}$. - Division: $3^{4} \div 3^{n+3} = 3^{4 - (n+3)} = 3^{1 - n}$. (xi) Simplify $\frac{81 \times 27^{2n}}{3^{5n+1}}$: - Express bases as powers of 3: $81=3^4$, $27=3^3$. - Numerator: $3^4 \times (3^3)^{2n} = 3^4 \times 3^{6n} = 3^{4+6n}$. - Divide by denominator: $3^{4+6n - (5n+1)} = 3^{n+3}$. (xii) Simplify $\frac{32^{n+2}}{2^n 4^{n+2}}$: - Express bases as powers of 2: $32=2^5$, $4=2^2$. - Numerator: $(2^5)^{n+2} = 2^{5n + 10}$. - Denominator: $2^n \times (2^2)^{n+2} = 2^n \times 2^{2n + 4} = 2^{3n + 4}$. - Divide: $2^{5n + 10 - (3n + 4)} = 2^{2n + 6}$. 3. Solve the following equations for the unknown. (i) Solve $4^{3n} = 32$: - Express bases as powers of 2: $4=2^2$, $32=2^5$. - Equation: $(2^2)^{3n} = 2^5 \Rightarrow 2^{6n} = 2^5$. - Equate exponents: $6n = 5 \Rightarrow n = \frac{5}{6}$. (ii) Solve $3^{7n - 15} = (\frac{1}{3})^{y + 3}$: - Rewrite right side: $(\frac{1}{3})^{y+3} = 3^{-(y+3)}$. - Equate exponents: $7n - 15 = -(y + 3)$. - Rearranged: $7n - 15 = -y - 3$. (iii) Solve $4^{3n + 6} = 1$: - $1 = 4^0$. - Equate exponents: $3n + 6 = 0 \Rightarrow n = -2$. (iv) Solve $4 \times 2^{6 - 3n} = 32$: - Express 4 and 32 as powers of 2: $4=2^2$, $32=2^5$. - Equation: $2^2 \times 2^{6 - 3n} = 2^5 \Rightarrow 2^{2 + 6 - 3n} = 2^5$. - Exponents: $8 - 3n = 5 \Rightarrow -3n = -3 \Rightarrow n = 1$. (v) Solve $8^{3x - 9} - 12 = 500$: - Add 12: $8^{3x - 9} = 512$. - Express 8 and 512 as powers of 2: $8=2^3$, $512=2^9$. - Equation: $(2^3)^{3x - 9} = 2^9 \Rightarrow 2^{9x - 27} = 2^9$. - Equate exponents: $9x - 27 = 9 \Rightarrow 9x = 36 \Rightarrow x = 4$. (vi) Solve $81^{x+1} \times 9^{1+4x} = 35^x$: - Express 81 and 9 as powers of 3: $81=3^4$, $9=3^2$. - Left side: $3^{4(x+1)} \times 3^{2(1+4x)} = 3^{4x + 4 + 2 + 8x} = 3^{12x + 6}$. - Right side: $35^x$ (cannot express as power of 3). - Equation: $3^{12x + 6} = 35^x$. - This equation cannot be solved with integer exponents; requires logarithms. (vii) Solve $10 \times 3^{x+1} = 243$: - Express 243 as $3^5$. - Divide both sides by 10: $3^{x+1} = \frac{243}{10}$. - Take logarithm base 3: $x + 1 = \log_3 \frac{243}{10}$. - $x = \log_3 \frac{243}{10} - 1$. (viii) Solve $2 \times 25^{x-1} - 7 = 243$: - Add 7: $2 \times 25^{x-1} = 250$. - Divide by 2: $25^{x-1} = 125$. - Express bases as powers of 5: $25=5^2$, $125=5^3$. - Equation: $(5^2)^{x-1} = 5^3 \Rightarrow 5^{2x - 2} = 5^3$. - Equate exponents: $2x - 2 = 3 \Rightarrow 2x = 5 \Rightarrow x = \frac{5}{2}$. (ix) Solve $(3.14)^{p-1} + 11 = 12$: - Subtract 11: $(3.14)^{p-1} = 1$. - Since any number to the power 0 is 1, $p - 1 = 0 \Rightarrow p = 1$. Final answers: (i) $4$ (ii) $2^{7n}$ (iii) $5a^8 b^{13}$ (iv) $y^8 z^{22}$ (v) $\frac{1}{324}$ (vi) $\frac{1}{8}$ (vii) $\frac{15}{x^3}$ (viii) $\frac{1}{3^7 x^6}$ (ix) $8 a^4 b^5$ (x) $5^{2n + 3}$ (xi) $\frac{3}{\sqrt{14}} a^2$ (xii) $20$ 2(i) $2$ 2(ii) $27$ 2(iii) $\frac{1}{8}$ 2(iv) $\frac{1}{343}$ 2(v) $5^{10/3}$ 2(vi) $\frac{3}{2}$ 2(vii) $5^2 = 25$ 2(viii) $\frac{9}{\sqrt{5}} 2^{-50/3}$ 2(ix) $2^{n+2} 3^9$ 2(x) $3^{1-n}$ 2(xi) $3^{n+3}$ 2(xii) $2^{2n + 6}$ 3(i) $n=\frac{5}{6}$ 3(ii) $7n - 15 = -(y + 3)$ 3(iii) $n = -2$ 3(iv) $n=1$ 3(v) $x=4$ 3(vi) No simple solution without logarithms 3(vii) $x = \log_3 \frac{243}{10} - 1$ 3(viii) $x=\frac{5}{2}$ 3(ix) $p=1$