Question 1. a Simplify (81/256)^-3/4 b x^m+2n. x3m-6n c If n = 3^(2/3) + 3^(-2/3), show that 9n2 - 27n = 82

let's tackle these exponent and algebra problems. a) To simplify (81/256)^-3/4: Step 1: Apply the negative exponent rule, which states a^-n = (1)/(a^n) or ((a)/(b))^-n = ((b)/(a))^n. ((81)/(256))^-3/4 = ((256)/(81))^3/4 Step 2: Express the base numbers as powers. 256 = 4^4 and 81 = 3^4. ((256)/(81))^3/4 = ((4^4)/(3^4))^3/4 = (((4)/(3))^4)^3/4 Step 3: Apply the power of a power rule, (a^m)^n = a^mn. (((4)/(3))^4)^3/4 = ((4)/(3))^4 · (3)/(4) = ((4)/(3))^3 Step 4: Calculate the final power. ((4)/(3))^3 = (4^3)/(3^3) = (64)/(27) The simplified expression is (64)/(27). b) To simplify x^m+2n · x^3m-8nx^5m-6n: Step 1: Apply the product rule for exponents in the numerator, a^p · a^q = a^p+q. x^m+2n · x^3m-8n = x^(m+2n) + (3m-8n) = x^m+3m+2n-8n = x^4m-6n Step 2: Apply the quotient rule for exponents, (a^p)/(a^q) = a^p-q. x^4m-6nx^5m-6n = x^(4m-6n) - (5m-6n) Step 3: Simplify the exponent. x^4m-6n - 5m + 6n = x^4m-5m - 6n+6n = x^-m Step 4: Express with a positive exponent. x^-m = (1)/(x^m) The simplified expression is (1)/(x^m). c) If n = 3^2/3 + 3^-2/3, show that 9n^2 - 27n = 82. Note: There appears to be a typo in the question. Based on the common structure of such problems and the derivation below, the equation to be shown is likely 9n^3 - 27n = 82, not 9n^2 - 27n = 82. Also, the variable in the equation should be n, not x. I will proceed with showing 9n^3 - 27n = 82 assuming these corrections. Step 1: Start with the given expression for n. n = 3^2/3 + 3^-2/3 Step 2: Cube both sides of the equation. Recall the identity (a+b)^3 = a^3 + b^3 + 3ab(a+b). Let a = 3^2/3 and b = 3^-2/3. n^3 = (3^2/3)^3 + (3^-2/3)^3 + 3(3^2/3)(3^-2/3)(3^2/3 + 3^-2/3) Step 3: Simplify each term. (3^2/3)^3 = 3^(2/3) · 3 = 3^2 = 9 (3^-2/3)^3 = 3^(-2/3) · 3 = 3^-2 = (1)/(9) 3(3^2/3)(3^-2/3) = 3 · 3^(2/3 - 2/3) = 3 · 3^0 = 3 · 1 = 3 The term (3^2/3 + 3^-2/3) is equal to n. Step 4: Substitute the simplified terms back into the equation for n^3. n^3 = 9 + (1)/(9) + 3n Step 5: Rearrange the equation to isolate the terms involving n. n^3 - 3n = 9 + (1)/(9) n^3 - 3n = (81)/(9) + (1)/(9) n^3 - 3n = (82)/(9) Step 6: Multiply the entire equation by 9 to clear the denominator. 9(n^3 - 3n) = 9 ((82)/(9)) 9n^3 - 27n = 82 Proven (assuming typo corrected to 9n^3 - 27n = 82) What's next?