This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To simplify the expression [ ( (16)/(9) )^-(3)/(2) × 16^-(3)/(4) ]^(1)/(3), we will simplify the terms inside the bracket first, then apply the outer exponent. Step 1: Simplify the first term inside the bracket, ( (16)/(9) )^-(3)/(2). Using the exponent rule a^-n = (1)/(a^n) and ((a)/(b))^-n = ((b)/(a))^n: ( (16)/(9) )^-(3)/(2) = ( (9)/(16) )^(3)/(2) Now, using the rule a^(m)/(n) = ([n]a)^m: ( (9)/(16) )^(3)/(2) = ( sqrt((9)/(16)) )^3 = ( sqrt(9)sqrt(16) )^3 = ( (3)/(4) )^3 ( (3)/(4) )^3 = (3^3)/(4^3) = (27)/(64) Step 2: Simplify the second term inside the bracket, 16^-(3)/(4). Using the exponent rule a^-n = (1)/(a^n): 16^-(3)/(4) = (1)/(16^3)4 Now, using the rule a^(m)/(n) = ([n]a)^m: (1)/(16^3)4 = (1)/(([4]16))^3 Since 2^4 = 16, [4]16 = 2: (1)/((2)^3) = (1)/(8) Step 3: Multiply the simplified terms inside the bracket. (27)/(64) × (1)/(8) = (27 × 1)/(64 × 8) = (27)/(512) Step 4: Apply the outer exponent (1)/(3) to the result. ( (27)/(512) )^(1)/(3) Using the rule a^(1)/(n) = [n]a: ( (27)/(512) )^(1)/(3) = 3/(512) = [3]27[3]512 Since 3^3 = 27 and 8^3 = 512: [3]27[3]512 = (3)/(8) The simplified expression is (3)/(8). Comparing this with the given options: A. (3)/(4) B. (9)/(16) C. (3)/(8) D. (1)/(4) The correct option is C. The final answer is C. (3)/(8). Send me the next one 📸