Q: what is standard form of 3x3x3x3

The correct answer is: $81$ Explanation This is repeated multiplication of 3 four times, which can be written using an exponent as $3^4$. Calculate step by step: Steps: First step: $$3\times3=9$$ Second step: $$9\times3=27$$ Final step: $$27\times3=81$$ Therefore, $3^4 = 81$.

Q: What is the formula to find the total surface area?

The formula depends on the 3D shape. Here are the total surface area (TSA) formulas for common solids: Cube: TSA = $6a^2$ (where $a$ is the edge length) Rectangular prism (box): TSA = $2(ab + bc + ac)$ (edges $a,b,c$) Sphere: TSA = $4\pi r^2$ (radius $r$) Right circular cylinder: TSA = $2\pi r^2 + 2\pi r h$ (two circular bases plus curved surface; radius $r$, height $h$) Right circular cone: TSA = $\pi r^2 + \pi r l$ (base area $+\,$lateral area; $r$ = base radius, $l$ = slant height) Right prism (general): TSA = $2B + Ph$ (where $B$ = area of one base, $P$ = perimeter of the base, $h$ = prism height) Pyramid (regular or not): TSA = $B + \tfrac{1}{2}Pl$ (base area $B$, perimeter of base $P$, slant height $l$) Hemisphere: curved surface = $2\pi r^2$; including base (flat face) TSA = $3\pi r^2$. If you meant a specific solid, tell me which one and I’ll show a worked example.

Q: 30% of 900

The correct answer is: $270$ Explanation Thirty percent means thirty per hundred, so 30% of a number equals 0.3 times that number. Multiplying 0.3 by 900 gives the result. Steps: First step: $$30\% = \frac{30}{100} = 0.3$$ Second step: $$0.3 \times 900 = 270$$ Final step: $$\text{Therefore, }30\%\text{ of }900\text{ is }270.$$ Therefore, 30% of 900 is 270.

Q: What is 5 over 6 divided by 8?

The correct answer is: $\frac{5}{48}$ Explanation You divide by 8 by multiplying by its reciprocal (1/8). That gives a product of two fractions which you multiply across numerator and denominator. Steps: First step: $$\frac{5}{6}\div 8 = \frac{5}{6}\times\frac{1}{8}$$ Second step: $$\frac{5\times1}{6\times8} = \frac{5}{48}$$ Final step: $$\frac{5}{48}\text{ (cannot be simplified further)}$$ Therefore, $ \frac{5}{6} $ divided by $ 8 $ equals $ \frac{5}{48} $ (approximately $0.1041667$).

Q: Why is the value of k taken as 9×10

The short answer: Because $k$ is Coulomb's constant, defined by $k=\dfrac{1}{4\pi\varepsilon_0}$, and using the measured permittivity of free space gives $k\approx8.99\times10^9$, which is rounded to $9\times10^9$ for simplicity. Explanation Coulomb's law for the electrostatic force between two point charges is usually written as $$F=k\frac{q_1q_2}{r^2}.$$ The constant $k$ is related to the permittivity of free space $\varepsilon_0$ by $$k=\frac{1}{4\pi\varepsilon_0}.$$ The measured value of the vacuum permittivity is $\varepsilon_0\approx8.854187817\times10^{-12}\text{ F/m}.$ Plugging this into the formula gives $$k=\frac{1}{4\pi(8.854187817\times10^{-12})}\approx8.98755179\times10^9\text{ N·m}^2\text{/C}^2.$$ For most textbook and classroom work this is rounded to a convenient value: $$k\approx9\times10^9\text{ N·m}^2\text{/C}^2.$$ The rounding makes calculations easier and the small difference (about 0.14%) is negligible in typical problems. Why that particular number? The numerical value comes from the chosen SI units (meter, kilogram, second, ampere) and the experimentally measured $\varepsilon_0$. Conceptually, $\varepsilon_0$ and hence $k$ arise from the way the electric field behaves in free space (Maxwell’s equations). The value is not arbitrary — it is determined by measurements and the SI system. If you’d like, I can show the numeric substitution step-by-step or explain how the value connects to other constants (e.g., the speed of light and magnetic constant).

Question

When do we use R constant as 0.0821 and 8.

Accepted Answer

Use R = 0.0821 when pressure is in atmospheres and volume in liters; use R = 8.314 when you work in SI/energy units (pressure in Pa or kPa and volume in m³, or when R must be in joules).

Explanation

The gas constant appears in the ideal gas law $PV = nRT$. You must choose the R value whose units match the units of $P$ and $V$ you use.
Common forms:
$R = 0.082057\ 	ext{L·atm·mol}^{-1}	ext{·K}^{-1}$ (often rounded to 0.0821) — use if $P$ is in atm and $V$ is in L.
$R = 8.314462\ 	ext{J·mol}^{-1}	ext{·K}^{-1}$ (often rounded to 8.314 or 8.31) — use if you need energy units (J) or if $P$ is in Pa and $V$ in m³ (SI).
Other forms: $R = 8.314\ 	ext{kPa·L·mol}^{-1}	ext{·K}^{-1}$ or $0.08314\ 	ext{bar·L·mol}^{-1}	ext{·K}^{-1}$, etc.

You can convert between them because they are the same constant expressed with different unit combinations:
$$8.314\text{ J·mol}^{-1}\text{·K}^{-1}=8.314\text{ kPa·L·mol}^{-1}\text{·K}^{-1}=0.082057\text{ L·atm·mol}^{-1}\text{·K}^{-1}.$$

Quick rule of thumb

Use 0.0821 when PV is in (atm·L).
Use 8.314 when PV is in (J) or when P is in Pa (or kPa) and V in m³ (or convert to kPa·L).

Always keep units consistent; otherwise your numeric answer will be wrong.

Short example
Find n from $PV = nRT$ with $P=2.00\ 	ext{atm}$, $V=10.0\ 	ext{L}$, $T=300\ 	ext{K}$:$$n=\frac{PV}{RT}=\frac{2.00	imes10.0}{0.0821	imes300}\approx0.81\ 	ext{mol}.$$