answer:Let f(x) be the power function, which we can write as f(x) = x^alpha. Since the graph of the power function passes through point Aleft(frac {1}{2}, frac {sqrt {2}}{2}right), we have: left(frac {1}{2}right)^{alpha}= frac {sqrt {2}}{2} From this equation, we can deduce that: alpha= frac {1}{2} Therefore, the function can be expressed as: f(x) = x^{frac {1}{2}} Now, we find the derivative of f(x): f'(x) = frac {1}{2}x^{-frac {1}{2}} = frac {1}{2sqrt{x}} The slope of the tangent line at point A is given by: f'left(frac {1}{2}right) = frac {1}{2sqrt{frac {1}{2}}} = frac {sqrt {2}}{2} Since the tangent line passes through point Aleft(frac {1}{2}, frac {sqrt {2}}{2}right) and has a slope of frac {sqrt {2}}{2}, the point-slope form of the equation of the tangent line is: y - frac {sqrt {2}}{2} = frac {sqrt {2}}{2} left(x - frac {1}{2}right) Expanding this and rearranging terms gives us the equation of the tangent line in the general form: 2{sqrt {2}x - 4y + sqrt {2} = 0} So, the equation of the tangent line at point A is: boxed{2{sqrt {2}}x - 4y + {sqrt {2}} = 0} We started by expressing the power function and then, using the given point, found the expression for our function. Using the geometrical meaning of the derivative, we calculated the derivative of f(x) at x = frac {1}{2} to find the slope of the tangent line, and then we used point-slope form to write the equation of the tangent line, which we transformed to the general form. This problem evaluates the understanding of the definition of power functions and the geometrical significance of derivatives, as well as the ability to calculate and solve using these concepts. It is a foundational problem.

answer:Given that the original quadratic expression px^2 + qx + r = 5(x + 3)^2 - 15. Multiplying each term by 4, we obtain: [ 4px^2 + 4qx + 4r = 4[5(x + 3)^2 - 15] ] [ 4px^2 + 4qx + 4r = 20(x + 3)^2 - 60. ] We need to express 20(x + 3)^2 - 60 in the form m(x - h)^2 + k. Noticing that 20(x + 3)^2 - 60 is already in the desired form where m = 20, and the square term (x + 3)^2 indicates the vertex h = -3. Therefore, h = -3 remains the same after the transformation, thus: [ h = boxed{-3}. ]

answer:1. **Evaluate the Numerator**: [ 3^2 + 3^1 + 3^0 = 9 + 3 + 1 = 13 ] 2. **Evaluate the Denominator**: [ 3^{-1} + 3^{-2} + 3^{-3} = frac{1}{3} + frac{1}{9} + frac{1}{27} ] To add these fractions, find a common denominator, which is 27: [ frac{9}{27} + frac{3}{27} + frac{1}{27} = frac{13}{27} ] 3. **Divide the Numerator by the Denominator**: [ frac{13}{frac{13}{27}} = 13 times frac{27}{13} = 27 ] 4. **Conclusion**: After evaluating the expression, the result is 27, which will be 27. The final answer is boxed{(C) 27}

answer:Given that point P(-2,3) is symmetric with respect to the y-axis to point Q(a,b), we need to find the value of a+b. Step 1: Understand the symmetry with respect to the y-axis. This means that the x-coordinate of the point will change its sign, while the y-coordinate will remain the same. Step 2: Apply this understanding to point P(-2,3). Since P is symmetric to Q with respect to the y-axis, we have: - For the x-coordinate: a = -(-2) = 2 - For the y-coordinate: b = 3 Step 3: Calculate a+b: [a + b = 2 + 3 = 5] Therefore, the value of a+b is boxed{5}, which corresponds to option boxed{C}.