1. Simplify each expression and express with positive exponents. **a)** Simplify $\left(\frac{a^3}{b^{-4}}\right)^{-2}$ Step 1: Rewrite the expression inside the parentheses: $$\frac{a^3}{b^{-4}} = a^3 \times b^4$$ Step 2: Apply the outer exponent $-2$: $$\left(a^3 b^4\right)^{-2} = a^{3 \times (-2)} b^{4 \times (-2)} = a^{-6} b^{-8}$$ Step 3: Express with positive exponents by taking reciprocals: $$a^{-6} b^{-8} = \frac{1}{a^6 b^8}$$ **Answer a:** $$\frac{1}{a^6 b^8}$$ **b)** Simplify $$\frac{-12 x^{-3} (x y^2)^{-1}}{(2 x^3 y^{-1})^3}$$ Step 1: Simplify numerator: $$(x y^2)^{-1} = x^{-1} y^{-2}$$ So numerator: $$-12 x^{-3} \times x^{-1} y^{-2} = -12 x^{-4} y^{-2}$$ Step 2: Simplify denominator: $$(2 x^3 y^{-1})^3 = 2^3 x^{3 \times 3} y^{-1 \times 3} = 8 x^9 y^{-3}$$ Step 3: Write full fraction: $$\frac{-12 x^{-4} y^{-2}}{8 x^9 y^{-3}}$$ Step 4: Divide coefficients and variables: $$\frac{-12}{8} = -\frac{3}{2}$$ For $x$: $$x^{-4} / x^9 = x^{-4 - 9} = x^{-13}$$ For $y$: $$y^{-2} / y^{-3} = y^{-2 - (-3)} = y^{1}$$ Step 5: Combine: $$-\frac{3}{2} x^{-13} y^{1} = -\frac{3 y}{2 x^{13}}$$ **Answer b:** $$-\frac{3 y}{2 x^{13}}$$ 2. Evaluate without calculator. **a)** Evaluate $9^{-3/2}$ Step 1: Rewrite 9 as $3^2$: $$9^{-3/2} = (3^2)^{-3/2} = 3^{2 \times (-3/2)} = 3^{-3}$$ Step 2: Simplify: $$3^{-3} = \frac{1}{3^3} = \frac{1}{27}$$ **Answer a:** $\frac{1}{27}$ **b)** Evaluate $64^{1/2} \times 27^{-1/3}$ Step 1: Rewrite bases: $$64 = 2^6, \quad 27 = 3^3$$ Step 2: Apply exponents: $$64^{1/2} = (2^6)^{1/2} = 2^{6 \times 1/2} = 2^3 = 8$$ $$27^{-1/3} = (3^3)^{-1/3} = 3^{3 \times (-1/3)} = 3^{-1} = \frac{1}{3}$$ Step 3: Multiply: $$8 \times \frac{1}{3} = \frac{8}{3}$$ **Answer b:** $\frac{8}{3}$ 3. Solve for unknown. **a)** Solve $27^x = \frac{1}{9}$ Step 1: Rewrite bases as powers of 3: $$27 = 3^3, \quad 9 = 3^2$$ Step 2: Rewrite equation: $$3^{3x} = 3^{-2}$$ Step 3: Equate exponents: $$3x = -2$$ Step 4: Solve for $x$: $$x = -\frac{2}{3}$$ **Answer a:** $-\frac{2}{3}$ **b)** Solve $2^{x+4} + 2^x = 136$ Step 1: Factor out $2^x$: $$2^x (2^4 + 1) = 136$$ Step 2: Calculate inside parentheses: $$2^4 = 16$$ So: $$2^x (16 + 1) = 136$$ $$2^x \times 17 = 136$$ Step 3: Divide both sides by 17: $$2^x = \frac{136}{17} = 8$$ Step 4: Rewrite 8 as power of 2: $$8 = 2^3$$ Step 5: Equate exponents: $$x = 3$$ **Answer b:** $3$