Scientific Notation Conversions 4Ff0E9

1. **Problem:** Express each number in scientific notation. - Scientific notation expresses numbers as $$a \times 10^n$$ where $$1 \leq |a| < 10$$ and $$n$$ is an integer. 1. $$0.0000876 = 8.76 \times 10^{-5}$$ 2. $$1067000 = 1.067 \times 10^{6}$$ 3. $$12.508 = 1.2508 \times 10^{1}$$ 4. $$0.1760004 = 1.760004 \times 10^{-1}$$ 2. **Problem:** Rewrite each number in standard form. - Standard form means writing the number as a decimal number without exponents. 1. $$12.6 \times 10^{-3} = 0.0126$$ 2. $$0.0034 \times 10^{4} = 34$$ 3. $$0.009804 \times 10^{-6} = 9.804 \times 10^{-9} = 0.000000009804$$ 4. $$7.054 \times 10^{2} = 705.4$$ 3. **Problem:** Convert each measure to the equivalent metric unit. - Use metric prefixes and conversion factors. 1. $$0.045\, dam = 0.045 \times 10\, m = 0.45\, m = 45\, cm$$ 2. $$0.00175\, km^3 = 0.00175 \times (10^3)^3\, m^3 = 0.00175 \times 10^9\, m^3 = 1.75 \times 10^6\, m^3$$ Since $$1\, m^3 = 1000\, dm^3$$, then $$0.00175\, km^3 = 1.75 \times 10^6 \times 1000\, dm^3 = 1.75 \times 10^9\, dm^3$$ 3. $$23.110\, km = 23.110 \times 10^3\, m = 2.311 \times 10^4\, m$$ Since $$1\, m = 10^6\, \mu m$$, $$23.110\, km = 2.311 \times 10^4 \times 10^6 = 2.311 \times 10^{10}\, \mu m$$ 4. $$32\, cm^2 = 32 \times (10^{-2})^2\, m^2 = 32 \times 10^{-4}\, m^2 = 3.2 \times 10^{-3}\, m^2$$ Since $$1\, Gm^2 = (10^9)^2 = 10^{18}\, m^2$$, $$32\, cm^2 = 3.2 \times 10^{-3} \times 10^{-18} = 3.2 \times 10^{-21}\, Gm^2$$ 4. **Problem:** Convert each quantity to the indicated units. - Use appropriate conversion factors. 1. $$20534\, wks \to century$$ $$1\, century = 100\, years = 100 \times 52 = 5200\, weeks$$ $$20534 \div 5200 = 3.9496 \approx 3.95\, centuries$$ 2. $$2450\, bushels \to pint$$ $$1\, bushel = 32\, pints$$ $$2450 \times 32 = 78400\, pints$$ 3. $$9870000\, ml \to lb$$ $$1\, ml = 1\, g$$ (assuming water density) $$9870000\, ml = 9870000\, g = 9870\, kg$$ $$1\, lb = 0.453592\, kg$$ $$9870 \div 0.453592 = 21756.5\, lb$$ 4. $$54000\, days \to score$$ $$1\, score = 20\, years = 20 \times 365 = 7300\, days$$ $$54000 \div 7300 = 7.3973 \approx 7.4\, scores$$ 5. $$25.75\, miles \to inches$$ $$1\, mile = 5280\, ft = 5280 \times 12 = 63360\, inches$$ $$25.75 \times 63360 = 1,631,520\, inches$$ 6. $$2500\, gallons \to fl.oz.$$ $$1\, gallon = 128\, fl.oz.$$ $$2500 \times 128 = 320,000\, fl.oz.$$ 7. $$2.089\, oz \to qt$$ $$1\, qt = 32\, oz$$ $$2.089 \div 32 = 0.06528\, qt$$ 8. $$867.54\, grains \to lb$$ $$1\, lb = 7000\, grains$$ $$867.54 \div 7000 = 0.1239\, lb$$ 9. $$0.347\, pecks \to qt$$ $$1\, peck = 8\, qt$$ $$0.347 \times 8 = 2.776\, qt$$ 5. **Problem:** Express each ratio in simplest form. 1. $$18\, gal : 90\, pints$$ $$1\, gal = 8\, pints$$ $$18\, gal = 18 \times 8 = 144\, pints$$ Ratio: $$144 : 90 = \frac{144}{90} = \frac{16}{10} = \frac{8}{5}$$ Simplest form: $$8 : 5$$ 2. $$212 ^\circ F : 100 ^\circ C$$ This is a temperature comparison, not a ratio of quantities, so simplest form is the ratio as is. Alternatively, convert $$212 ^\circ F$$ to Celsius: $$C = \frac{5}{9}(F - 32) = \frac{5}{9}(212 - 32) = 100 ^\circ C$$ So ratio is $$100 : 100 = 1 : 1$$ 3. $$3\, lb : 48\, oz$$ $$1\, lb = 16\, oz$$ $$3\, lb = 3 \times 16 = 48\, oz$$ Ratio: $$48 : 48 = 1 : 1$$