answer:Given the equation: [ 2^{3x+1} - 17 cdot 2^{2x} + 2^{x+3} = 0 ] 1. **Simplify the equation by factoring out a common term**: We can factor out (2^x) from each term: [ 2^x left(2^{3x - x + 1} - 17 cdot 2^{2x - x} + 2^{x + 3 - x}right) = 0 ] Simplify the exponents: [ 2^x (2^{2x+1} - 17 cdot 2^x + 2^3) = 0 ] Which simplifies to: [ 2^x (2 cdot 2^{2x} - 17 cdot 2^x + 8) = 0 ] 2. **Introduce a substitution to solve the quadratic equation**: Let (y = 2^x). Then the equation transforms to: [ 2 cdot y^2 - 17 cdot y + 8 = 0 ] This is a standard quadratic equation (a y^2 + b y + c = 0). 3. **Solve the quadratic equation using the quadratic formula**: The quadratic formula is: [ y = frac{-b pm sqrt{b^2 - 4ac}}{2a} ] For our quadratic equation (2y^2 - 17y + 8 = 0): - (a = 2) - (b = -17) - (c = 8) Substitute these values into the formula: [ y = frac{-(-17) pm sqrt{(-17)^2 - 4 cdot 2 cdot 8}}{2 cdot 2} ] Simplify inside the square root: [ y = frac{17 pm sqrt{289 - 64}}{4} ] Further simplify: [ y = frac{17 pm sqrt{225}}{4} ] Since (sqrt{225} = 15): [ y = frac{17 pm 15}{4} ] This gives two solutions: [ y = frac{17 + 15}{4} = 8 quad text{and} quad y = frac{17 - 15}{4} = frac{1}{2} ] 4. **Find the corresponding (x) values**: Recall that (y = 2^x). Thus we have: [ 2^x = 8 quad Rightarrow quad x = log_2(8) = 3 ] And: [ 2^x = frac{1}{2} quad Rightarrow quad x = log_2left(frac{1}{2}right) = -1 ] 5. **Compute the product of the solutions**: We need the product of (x = 3) and (x = -1): [ 3 cdot (-1) = -3 ] Conclusion: [ boxed{-3} ]

answer:Starting with the equation: [ sqrt{a - sqrt{a + b^x}} = x ] first square both sides: [ a - sqrt{a + b^x} = x^2 ] From this, rearrange to get: [ a = x^2 + sqrt{a + b^x} ] This means: [ sqrt{a + b^x} = a - x^2 ] Square both sides again: [ a + b^x = (a - x^2)^2 ] This simplifies and expands to: [ b^x = a^2 - 2ax^2 + x^4 - a ] Considering the complexity, let’s return to the substitution using a new variable, leading to: [ b^x = frac{a - x^2 - x}{2x}, quad text{via the arrangement } sqrt{a+b^x} = frac{a - x^2 - x}{2x} ] Reverting back and simplifying the isolated representation: [ b^x = (a-x^2),(1 + frac{x}{2x}) = a-x^2 ] Now considering the exponential form with x = frac{-1 pm sqrt{4a - 3}}{2} based on solving x^2 + x + 1 - a = 0, we find: [ x = frac{-1 + sqrt{4a-3}}{2} ] As it remains the only solution when x geq 0 and falls back to satisfy the original equation, it translates to: [ frac{sqrt{4a - 3b - 1}{2}} ] The correct answer is boxed{E}.

answer:The correct answer is boxed{text{B}}. Brief explanation.

answer:To solve this expression, we need to follow the order of operations, which is Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This is often remembered by the acronym PEMDAS. First, we perform the multiplication: 34.2735 * 18.9251 = 648.5603855 Next, we perform the division: 648.5603855 / 6.8307 ≈ 94.9511 Then, we perform the addition: 94.9511 + 128.0021 = 222.9532 Finally, we perform the subtraction: 222.9532 - 56.1193 = 166.8339 So the result of the expression is approximately boxed{166.8339} .