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
Welcome back — been a while! Let's pick up where you left off. Step 1: Rewrite the exponent on the left side of the equation. The given equation is: 5^2x+6 = 7^x+3 Notice that 2x+6 can be factored as 2(x+3). So, the equation becomes: 5^2(x+3) = 7^x+3 Step 2: Apply the exponent rule (a^b)^c = a^bc. (5^2)^x+3 = 7^x+3 25^x+3 = 7^x+3 Step 3: Rearrange the equation to isolate the terms with x+3 in the exponent. Divide both sides by 7^x+3: 25^x+37^x+3 = 1 Step 4: Apply the exponent rule (a^c)/(b^c) = ((a)/(b))^c. ((25)/(7))^x+3 = 1 Step 5: Solve for x. For any non-zero base B, if B^A = 1, then the exponent A must be 0. In this case, B = (25)/(7) and A = x+3. Therefore, we must have: x+3 = 0 x = -3 Step 6: Verify the solution. Substitute x=-3 back into the original equation: Left side: 5^2(-3)+6 = 5^-6+6 = 5^0 = 1 Right side: 7^(-3)+3 = 7^0 = 1 Since 1=1, the solution is correct. The final answer is -3. Send me the next one 📸