(ii) \( \frac{81^{a x} \times 27^{x}}{9^{x}}=729 \) (iii) Find the value of \( x \) and \( y \). \( 9^{a y} \times 2^{x}=72 \) (iv) \( 3^{2 y}=81 \) Evaluate the following index expressions (i) \( 25^{3 / 2} \times 9^{3 / 3 / 2} \) in

Let's solve the problems step by step. ### Problem (ii) We have the equation: \[ \frac{81^{a x} \times 27^{x}}{9^{x}}=729 \] First, we can express all the numbers in terms of base 3: - \( 81 = 3^4 \) - \( 27 = 3^3 \) - \( 9 = 3^2 \) - \( 729 = 3^6 \) Now, substituting these into the equation gives: \[ \frac{(3^4)^{a x} \times (3^3)^{x}}{(3^2)^{x}} = 3^6 \] This simplifies to: \[ \frac{3^{4ax} \times 3^{3x}}{3^{2x}} = 3^6 \] Combining the exponents in the numerator: \[ \frac{3^{4ax + 3x}}{3^{2x}} = 3^6 \] Now, we can simplify the left side: \[ 3^{4ax + 3x - 2x} = 3^6 \] \[ 3^{4ax + x} = 3^6 \] Since the bases are the same, we can set the exponents equal to each other: \[ 4ax + x = 6 \] Factoring out \( x \): \[ x(4a + 1) = 6 \] Thus, we have: \[ x = \frac{6}{4a + 1} \] ### Problem (iii) We need to solve the equation: \[ 9^{a y} \times 2^{x} = 72 \] Again, express \( 9 \) and \( 72 \) in terms of base 3: - \( 9 = 3^2 \) - \( 72 = 8 \times 9 = 2^3 \times 3^2 \) Substituting gives: \[ (3^2)^{a y} \times 2^{x} = 2^3 \times 3^2 \] This simplifies to: \[ 3^{2ay} \times 2^{x} = 2^3 \times 3^2 \] Now, we can equate the bases: 1. For base 3: \[ 2ay = 2 \implies ay = 1 \implies y = \frac{1}{a} \] 2. For base 2: \[ x = 3 \] ### Problem (iv) We have the equation: \[ 3^{2y} = 81 \] Expressing \( 81 \) in terms of base 3: \[ 3^{2y} = 3^4 \] Setting the exponents equal gives: \[ 2y = 4 \implies y = 2 \] ### Summary of Results From the above calculations: - \( x = 3 \) - \( y = 2 \) - \( a = \frac{1}{2} \) (from \( ay = 1 \)) ### Problem (i) Now, we evaluate the expression: \[ 25^{3/2} \times 9^{3/(3/2)} \] First, simplify \( 9^{3/(3/2)} \): \[ 9^{3/(3/2)} = 9^{3 \cdot \frac{2}{3}} = 9^2 = 81 \] Now, simplify \( 25^{3/2} \): \[ 25^{3/2} = (5^2)^{3/2} = 5^{2 \cdot \frac{3}{2}} = 5^3 = 125 \] Now, multiply the two results: \[ 125 \times 81 \] Calculating \( 125 \times 81 \): \[ 125 \times 81 = 10125 \] Thus, the final result for the index expression is: \[ \boxed{10125} \]