4. Find the value of \( x \) \( \begin{array}{lll}\text { (a) }\left(\frac{5}{6}\right)^{3} \times\left(\frac{5}{6}\right)^{5}=\left(\frac{5}{6}\right)^{2 x} & \text { (b) }\left(\frac{7}{9}\right)^{-9} \times\left(\frac{7}{9}\right)^{6}=\left(\frac{7}{9}\right)^{2 x-1} & \text { (c) }\left(5^{-x} \times 4^{-x}\right)=20^{-1} \\ \text { (d) } x \times\left(4^{-2}\right)=\frac{1}{4} & \text { (e) } x(-3)^{3} \div x^{2}=-9 & \text { (f) }\left(\frac{1}{3}\right)^{3} \div\left(\frac{2}{3}\right)^{3}=\left(\frac{1}{2}\right)^{x}\end{array} \)

We will solve each part step‐by‐step. ───────────────────────────── (a) (5/6)³ × (5/6)⁵ = (5/6)^(3+5) = (5/6)⁸   This equals (5/6)^(2x)   Thus equate exponents: 8 = 2x, so x = 4. ───────────────────────────── (b) (7/9)^(–9) × (7/9)⁶ = (7/9)^(–9+6) = (7/9)^(–3)   This equals (7/9)^(2x–1)   Equate exponents: –3 = 2x – 1   Add 1 to both sides: –2 = 2x   So, x = –1. ───────────────────────────── (c) 5^(–x) × 4^(–x) = (5×4)^(–x) = 20^(–x)   This is equal to 20^(–1)   Thus, –x = –1, so x = 1. ───────────────────────────── (d) x × 4^(–2) = x × (1/4²) = x/16   We have: x/16 = 1/4   Multiply both sides by 16: x = 16/4 = 4. ───────────────────────────── (e) x · (–3)³ ÷ x² = x·(–27) ÷ x² = –27/x   Set up the equation: –27/x = –9   Multiply both sides by x: –27 = –9x   Divide by –9: x = 3. ───────────────────────────── (f) (1/3)³ ÷ (2/3)³ = [(1/3)/(2/3)]³ = (1/2)³   This equals (1/2)^(x)   Then equate exponents: 3 = x, so x = 3. ───────────────────────────── Final Answers:  (a) x = 4  (b) x = –1  (c) x = 1  (d) x = 4  (e) x = 3  (f) x = 3