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
Step 1: Simplify the right side of the equation using the exponent rule a^m · a^n = a^m+n. x^x = 2^x · 2^320 x^x = 2^x+320 Step 2: Divide both sides by 2^x. (x^x)/(2^x) = 2^x+3202^x Step 3: Simplify both sides using the exponent rules (a^m)/(b^m) = ((a)/(b))^m and (a^m)/(a^n) = a^m-n. ((x)/(2))^x = 2^x+320-x ((x)/(2))^x = 2^320 Step 4: To make the base and exponent on the left side equal, raise both sides to the power of (1)/(2). This uses the rule (a^b)^c = a^bc. (((x)/(2))^x)^(1)/(2) = (2^320)^(1)/(2) ((x)/(2))^(x)/(2) = 2^(320)/(2) ((x)/(2))^(x)/(2) = 2^160 Step 5: Express the right side in the form k^k. We need to find a number k such that k^k = 2^160. We can rewrite 2^160 as (2^a)^b where a · b = 160. We want b = 2^a. Let's try a=5. Then 2^a = 2^5 = 32. If a=5, then b = (160)/(5) = 32. So, 2^160 = (2^5)^32 = 32^32. The equation becomes: ((x)/(2))^(x)/(2) = 32^32 Step 6: Equate the bases (and exponents) since they are in the form A^A = B^B. (x)/(2) = 32 Step 7: Solve for x. x = 32 × 2 x = 64 The final answer is 64.