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's how to simplify the expressions: a) Simplify 4^2 × 4^-3 Step 1: Use the exponent rule a^m × a^n = a^m+n. 4^2 × 4^-3 = 4^2 + (-3) 4^2 + (-3) = 4^2-3 4^2-3 = 4^-1 Step 2: Convert the negative exponent to a positive exponent using the rule a^-n = (1)/(a^n). 4^-1 = (1)/(4^1) 4^-1 = (1)/(4) The simplified expression is (1)/(4). b) Simplify 3^2 (2^2)^-22^3 Step 1: Simplify the term (2^2)^-2 in the numerator using the exponent rule (a^m)^n = a^mn. (2^2)^-2 = 2^2 × (-2) (2^2)^-2 = 2^-4 Step 2: Substitute this back into the expression. 3^2 × 2^-42^3 Step 3: Simplify the terms with base 2 using the exponent rule (a^m)/(a^n) = a^m-n. 2^-42^3 = 2^-4-3 2^-4-3 = 2^-7 Step 4: Combine the simplified terms. 3^2 × 2^-7 Step 5: Convert the negative exponent to a positive exponent using the rule a^-n = (1)/(a^n). 2^-7 = (1)/(2^7) Step 6: Write the expression with positive exponents. 3^2 × (1)/(2^7) = (3^2)/(2^7) Step 7: Calculate the values of the powers. 3^2 = 9 2^7 = 128 The simplified expression is (9)/(128).