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question:How do you simplify the expression (9^2 times 3^{-2})?
answer:To simplify the expression (9^2 times 3^{-2}), follow these steps: 1. Calculate (9^2), which equals 81. 2. Handle the negative exponent (3^{-2}). When an exponent is negative, it means you take the reciprocal of the base and then raise it to the absolute value of the exponent. So, (3^{-2} = frac{1}{3^2} = frac{1}{9}). 3. Multiply the results from steps 1 and 2: (81 times frac{1}{9} = 9). Thus, the simplified form of (9^2 times 3^{-2}) is 9.
question:During an expansion (or recovery), the percent change in which of the following economic indicators will be positive? a) Real GDP b) Real capita GDP c) Consumer Price Index (CPI) d) GDP Deflator
answer:The correct answer is a. Real GDP. In an economic expansion or recovery, real GDP, which represents the value of all final goods and services produced within a country adjusted for inflation, typically experiences a positive growth rate, indicating an increase in the level of output.
question:A flotation device with a density of 0.6 times that of water is shaped like a right cylinder. It has a height of 0.5 m and a face area of 1.5 m<sup>2</sup>. If the device is fully submerged in water, what is the magnitude of the buoyant force acting on it?
answer:The density of water is {eq}rho_w = 1000 rm kg/m^3{/eq}. The density of the flotation device is {eq}rho_c = 0.6rho_w = 600 rm kg/m^3{/eq}. The volume of the cylinder is {eq}V_c = Ah = 1.5 rm m^2 times 0.5 rm m = 0.75 rm m^3{/eq}. The buoyant force is {eq}F_B = rho_w g V_c = 1000 rm kg/m^3 times 9.81 rm m/s^2 times 0.75 rm m^3 = boxed{7357.5 rm N{/eq}.
question:Find the sum of the polynomials p(x) = 3(x^2 - 5x - 1) and q(x) = -14x^2 + x + 7.
answer:To find the sum of two polynomials, we simply add the like terms together. p(x) + q(x) = (3x^2 - 15x - 3) + (-14x^2 + x + 7) = 3x^2 - 15x - 3 - 14x^2 + x + 7 = (3x^2 - 14x^2) + (-15x + x) + (-3 + 7) = -11x^2 - 14x + 4 Therefore, the sum of the polynomials p(x) and q(x) is -11x^2 - 14x + 4. The answer is -11x^2 - 14x + 6