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: Solve the equation 49^x+1 + 7^2x = 350. Rewrite the terms with a common base of 7. (7^2)^x+1 + 7^2x = 350 7^2(x+1) + 7^2x = 350 7^2x+2 + 7^2x = 350 Apply the exponent rule a^m+n = a^m · a^n. 7^2x · 7^2 + 7^2x = 350 49 · 7^2x + 7^2x = 350 Factor out 7^2x. 7^2x(49 + 1) = 350 50 · 7^2x = 350 Divide by 50. 7^2x = (350)/(50) 7^2x = 7 Since the bases are the same, equate the exponents. 2x = 1 x = (1)/(2) The value of x is (1)/(2). Step 2: Solve the equation 16^x = 8^4x-3. Rewrite the terms with a common base of 2. (2^4)^x = (2^3)^4x-3 2^4x = 2^3(4x-3) 2^4x = 2^12x-9 Since the bases are the same, equate the exponents. 4x = 12x - 9 Subtract 4x from both sides. 0 = 8x - 9 Add 9 to both sides. 9 = 8x Divide by 8. x = (9)/(8) The value of x is (9)/(8). Step 3: Solve the equation ((1)/(27))^m × (81)^-1 = 243. Rewrite the terms with a common base of 3. (3^-3)^m × (3^4)^-1 = 3^5 Apply the exponent rule (a^b)^c = a^bc. 3^-3m × 3^-4 = 3^5 Apply the exponent rule a^m · a^n = a^m+n. 3^-3m-4 = 3^5 Since the bases are the same, equate the exponents. -3m - 4 = 5 Add 4 to both sides. -3m = 9 Divide by -3. m = (9)/(-3) m = -3 The value of m is -3. Step 4: Express 3^2y-1 + 2 × 3^y-1 = 1 in terms of P, given P = 3^y, and find the value of y. First, express the equation in terms of P. 3^2y-1 + 2 × 3^y-1 = 1 Rewrite the terms using 3^y and 3^-1. 3^2y · 3^-1 + 2 × 3^y · 3^-1 = 1 (3^y)^2 · (1)/(3) + 2 · 3^y · (1)/(3) = 1 Substitute P = 3^y. P^2 · (1)/(3) + 2 · P · (1)/(3) = 1 (P^2)/(3) + (2P)/(3) = 1 Multiply the entire equation by 3. P^2 + 2P = 3 Rearrange into a quadratic equation. P^2 + 2P - 3 = 0 The equation in terms of P is P^2 + 2P - 3 = 0. Next, find the value of y. Solve the quadratic equation for P. P^2 + 2P - 3 = 0 Factor the quadratic equation. (P+3)(P-1) = 0 This gives two possible values for P: P = -3 or P = 1 Recall that P = 3^y. Case 1: 3^y = -3. There is no real solution for y because 3^y is always positive. Case 2: 3^y = 1. Rewrite 1 as 3^0. 3^y = 3^0 Equate the exponents. y = 0 The value of y is 0. Step 5: Simplify sqrt(2^x × 5^2x) ÷ 2^x. Simplify the expression inside the square root first. 2^x × 5^2x ÷ 2^x = 2^x × 5^2x2^x Cancel out 2^x. = 5^2x Now, take the square root of the simplified expression. sqrt(5^2x) = (5^2x)^(1)/(2) Apply the exponent rule (a^b)^c = a^bc. = 5^2x × (1)/(2) = 5^x The simplified expression is 5^x.