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
Here are the solutions for the expressions: e) Simplify 2^n × (2^n)^3 × 2^2n Step 1: Simplify the term (2^n)^3 using the exponent rule (a^m)^n = a^mn. (2^n)^3 = 2^n × 3 = 2^3n Step 2: Substitute this back into the expression. 2^n × 2^3n × 2^2n Step 3: Combine the terms using the exponent rule a^m × a^n × a^p = a^m+n+p. 2^n × 2^3n × 2^2n = 2^n + 3n + 2n 2^n + 3n + 2n = 2^6n The simplified expression is 2^6n. f) Simplify 3^m × 27^m × 9^-m Step 1: Express all bases as powers of 3. 27 = 3^3 9 = 3^2 Step 2: Substitute these into the expression. 3^m × (3^3)^m × (3^2)^-m Step 3: Simplify the terms with powers of powers using (a^x)^y = a^xy. (3^3)^m = 3^3m (3^2)^-m = 3^2 × (-m) = 3^-2m Step 4: Substitute these back into the expression. 3^m × 3^3m × 3^-2m Step 5: Combine the terms using the exponent rule a^x × a^y × a^z = a^x+y+z. 3^m × 3^3m × 3^-2m = 3^m + 3m + (-2m) 3^m + 3m - 2m = 3^4m - 2m 3^4m - 2m = 3^2m The simplified expression is 3^2m. g) Simplify (a^1/2 × a)^2.5 Step 1: Simplify inside the parenthesis. Remember that a = a^1. Use the exponent rule a^x × a^y = a^x+y. a^1/2 × a^1 = a^1/2 + 1 a^1/2 + 1 = a^1/2 + 2/2 = a^3/2 Step 2: Substitute this back into the expression. (a^3/2)^2.5 Step 3: Convert the decimal exponent 2.5 to a fraction. 2.5 = (25)/(10) = (5)/(2) Step 4: Apply the outer exponent using the rule (a^x)^y = a^xy. (a^3/2)^5/2 = a^(3)/(2) × (5)/(2) a^(3)/(2) × (5)/(2) = a^(3 × 5)/(2 × 2) a^(15)/(4) The simplified expression is a^(15)/(4).