Question: Solve the following problems:
1. **Write as a power of 2:**
a. $\sqrt[4]{8}$
b. $\frac{1}{\sqrt[3]{16}}$
c. $\frac{4}{\sqrt{32}}$
2. **Write in standard form:**
a. 1700
b. 468 200
c. 0.058
d. 0.000 606
3. **Write as an ordinary number:**
a. $7.71 \times 10^{-3}$
b. $4 \times 10^{4}$
c. $1.702 \times 10^{-5}$
d. $5.8 \times 10^{7}$
4. **Write the diameter of Jupiter (139 800 km) in standard form.**
5. **Evaluate, giving your answer in standard form:**
a. $(4 \times 10^{6}) \times (7 \times 10^{4})$
b. $(3 \times 10^{-4})^{2}$
c. $\frac{3 \times 10^{8}}{6 \times 10^{8}}$
d. $(4 \times 10^{10}) + (2.5 \times 10^{11})$
6. **Find the total mass of bacteria in the drink:**
Number of bacteria = $6.5 \times 10^{8}$
Mass per bacterium = $4.2 \times 10^{-13}$ g
---
### Step-by-step solutions:
1. Write as a power of 2:
a. $\sqrt[4]{8} = 8^{\frac{1}{4}} = (2^{3})^{\frac{1}{4}} = 2^{\frac{3}{4}}$
b. $\frac{1}{\sqrt[3]{16}} = 16^{-\frac{1}{3}} = (2^{4})^{-\frac{1}{3}} = 2^{-\frac{4}{3}}$
c. $\frac{4}{\sqrt{32}} = 4 \times 32^{-\frac{1}{2}} = 2^{2} \times (2^{5})^{-\frac{1}{2}} = 2^{2} \times 2^{-\frac{5}{2}} = 2^{2 - \frac{5}{2}} = 2^{-\frac{1}{2}}$
2. Write in standard form:
a. $1700 = 1.7 \times 10^{3}$
b. $468200 = 4.682 \times 10^{5}$
c. $0.058 = 5.8 \times 10^{-2}$
d. $0.000606 = 6.06 \times 10^{-4}$
3. Write as an ordinary number:
a. $7.71 \times 10^{-3} = 0.00771$
b. $4 \times 10^{4} = 40000$
c. $1.702 \times 10^{-5} = 0.00001702$
d. $5.8 \times 10^{7} = 58000000$
4. Diameter of Jupiter in standard form:
$139800 = 1.398 \times 10^{5}$ km
5. Evaluate in standard form:
a. $(4 \times 10^{6}) \times (7 \times 10^{4}) = (4 \times 7) \times 10^{6+4} = 28 \times 10^{10} = 2.8 \times 10^{11}$
b. $(3 \times 10^{-4})^{2} = 3^{2} \times (10^{-4})^{2} = 9 \times 10^{-8}$
c. $\frac{3 \times 10^{8}}{6 \times 10^{8}} = \frac{3}{6} \times 10^{8-8} = \frac{1}{2} = 0.5$
d. $(4 \times 10^{10}) + (2.5 \times 10^{11}) = 0.4 \times 10^{11} + 2.5 \times 10^{11} = (0.4 + 2.5) \times 10^{11} = 2.9 \times 10^{11}$
6. Total mass of bacteria:
Total mass = Number of bacteria $\times$ Mass per bacterium
$= (6.5 \times 10^{8}) \times (4.2 \times 10^{-13}) = (6.5 \times 4.2) \times 10^{8 + (-13)} = 27.3 \times 10^{-5} = 2.73 \times 10^{-4}$ g
**Final answers:**
1a. $2^{\frac{3}{4}}$
1b. $2^{-\frac{4}{3}}$
1c. $2^{-\frac{1}{2}}$
2a. $1.7 \times 10^{3}$
2b. $4.682 \times 10^{5}$
2c. $5.8 \times 10^{-2}$
2d. $6.06 \times 10^{-4}$
3a. $0.00771$
3b. $40000$
3c. $0.00001702$
3d. $58000000$
4. $1.398 \times 10^{5}$ km
5a. $2.8 \times 10^{11}$
5b. $9 \times 10^{-8}$
5c. $0.5$
5d. $2.9 \times 10^{11}$
6. $2.73 \times 10^{-4}$ g