Scientific notation

Write the following decimal numbers in standard form:

1. a) $450 000 000$
2. b) $0,000000032$
3. c) $12 500$
4. d) $0,000105$

State which of the following numbers are not correctly written in standard form and write them correctly:

1. a) $45,2 \cdot 10^3$
2. b) $0,8 \cdot 10^{-2}$
3. c) $3,1 \cdot 10^5$
4. d) $12 \cdot 10^{-4}$

Express the following numbers given in standard form as ordinary decimals:

1. a) $3,4 \cdot 10^5$
2. b) $1,89 \cdot 10^{-4}$
3. c) $7 \cdot 10^8$
4. d) $5,02 \cdot 10^{-6}$

Determine the order of magnitude (the closest power of $10$) of the following quantities expressed in standard form:

1. a) $4,8 \cdot 10^6$
2. b) $1,1 \cdot 10^{-5}$
3. c) $9,2 \cdot 10^{12}$
4. d) $2,5 \cdot 10^{-8}$

Order the following numbers written in standard form from smallest to largest:

$A = 3,2 \cdot 10^4$

$B = 4,1 \cdot 10^{-3}$

$C = 1,5 \cdot 10^4$

$D = 8,9 \cdot 10^{-4}$

$E = 2 \cdot 10^5$

Perform the following multiplications by grouping the numbers and the powers of base $10$. Express the final result in standard form:

1. a) $(3 \cdot 10^4) \cdot (2 \cdot 10^5)$
2. b) $(4 \cdot 10^{-3}) \cdot (2 \cdot 10^{-4})$
3. c) $(5 \cdot 10^6) \cdot (3 \cdot 10^{-2})$

Calculate the following additions and subtractions with different exponents, first adjusting the powers of $10$ so that they match. Express the answer in standard form:

1. a) $3,5 \cdot 10^6 + 4,2 \cdot 10^5$
2. b) $7,1 \cdot 10^{-4} - 5 \cdot 10^{-5}$

Solve the following combined operation respecting the order of operations and give the final result in standard form:

$\frac{(4,2 \cdot 10^5) \cdot (2 \cdot 10^{-3})}{3 \cdot 10^4}$

The mean distance from the Earth to the Sun is about $1,5 \cdot 10^8$ km and the speed of light in a vacuum is $3 \cdot 10^5$ km/s. Using the formula for uniform motion in a straight line ($t = \frac{d}{v}$), how many seconds does it take for sunlight to reach the Earth? Express the answer in standard form and then convert it to minutes.

Set up and solve the following linear equation in which the coefficients are expressed in standard form. Rearrange for the unknown $x$ and express its value in standard form:

$(2,5 \cdot 10^4) \cdot x + 3 \cdot 10^5 = 1,8 \cdot 10^6$

Newton's Law of Universal Gravitation states that the gravitational force between two bodies is given by $F = G \frac{m_1 \cdot m_2}{d^2}$.

Knowing that the gravitational constant is $G = 6,67 \cdot 10^{-11}$ $N \cdot m^2/kg^2$, the mass of the Earth is $m_1 = 5,97 \cdot 10^{24}$ kg, the mass of the Moon is $m_2 = 7,35 \cdot 10^{22}$ kg and the mean distance between their centres is $d = 3,84 \cdot 10^8$ m, calculate the gravitational force $F$ in Newtons ($N$). Express the result in standard form rounded to two decimal places.

A glass contains exactly $18$ grams of pure water ($H_2O$). In chemistry, it is known that $18$ grams are equivalent to $1$ mol of water and that $1$ mol of any substance contains Avogadro's Number ($6,022 \cdot 10^{23}$) of molecules. From these data, calculate the mass in kilograms of a single molecule of water. Express the final result in standard form.