Scientific notation

Write the following numbers in standard form, adjusting the mantissa so that it lies between $1$ and $10$:

1. a) $0,000000456$
2. b) $34500000000$
3. c) $0,0023 \cdot 10^5$
4. d) $456 \cdot 10^{-4}$

Justify whether the following expressions are correctly written in standard form. If not, correct them so that the mantissa $N$ satisfies $1 \le |N| < 10$:

1. a) $12,4 \cdot 10^3$
2. b) $0,85 \cdot 10^{-2}$
3. c) $1 \cdot 10^6$
4. d) $-45,2 \cdot 10^{-8}$

The diameter of a human red blood cell is approximately $0,0075$ mm. Express this quantity in metres using standard form.

The estimated wealth of one of the richest people in the tech sector amounts to $185000$ million dollars. Write this figure in standard form.

Round the following numbers to three significant figures and then express them in standard form:

1. a) $4567892$
2. b) $0,00034091$
3. c) $98,99$

State the order of magnitude and the number of significant figures of the following experimental measurements:

1. a) $4,05 \cdot 10^8$ m
2. b) $1,200 \cdot 10^{-5}$ kg
3. c) $9 \cdot 10^{12}$ s

Order the following numbers from smallest to largest, first writing them in standard form and comparing their orders of magnitude:

$A = 0,0045 \cdot 10^{-3}$

$B = 450 \cdot 10^{-8}$

$C = 0,45 \cdot 10^{-5}$

$D = 45000 \cdot 10^{-11}$

Carry out the following multiplications and divisions mentally (without using a calculator), applying the index laws, and express the result in standard form:

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

Perform the following additions and subtractions of numbers in standard form that have the same exponent. Factorise out a common factor and give the final result:

1. a) $3,2 \cdot 10^5 + 4,5 \cdot 10^5$
2. b) $8,7 \cdot 10^{-4} - 3,2 \cdot 10^{-4}$
3. c) $5,15 \cdot 10^{12} + 2,03 \cdot 10^{12}$

Perform the following additions and subtractions, first adjusting the exponents so you can factorise out a common factor. Express the final result in standard form:

1. a) $4,5 \cdot 10^7 + 3,8 \cdot 10^8$
2. b) $9,2 \cdot 10^{-5} - 6,1 \cdot 10^{-6}$

Calculate the following mixed operation and express the final result in standard form, rounding to three significant figures:

$\frac{(2,5 \cdot 10^6) \cdot (4,2 \cdot 10^{-3})}{3,1 \cdot 10^{-4} + 1,9 \cdot 10^{-5}}$

The mass of the Earth is approximately $5,97 \cdot 10^{24}$ kg and that of the planet Jupiter is $1,90 \cdot 10^{27}$ kg. How many times more massive is Jupiter than the Earth? Express the result in standard form and also in standard decimal form.

The speed of light in a vacuum is approximately $3 \cdot 10^5$ km/s. A "light-year" is the distance light travels in a vacuum in one Earth year ($365,25$ days). Calculate how many kilometres a light-year corresponds to, and express the result in standard form to four significant figures.

Given that $18$ grams of water contain $6,022 \cdot 10^{23}$ molecules (Avogadro's number), calculate the mass in grams of a single molecule of water. Give the result in standard form to four significant figures.

Newton's law of universal gravitation states that the gravitational force of attraction between two masses is $F = G \frac{m_1 m_2}{d^2}$. Given that the constant $G = 6,67 \cdot 10^{-11}\text{ N}\cdot\text{m}^2/\text{kg}^2$, 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 $d = 3,84 \cdot 10^5$ km, calculate the force of attraction $F$ in newtons, using standard form in all intermediate calculations.