Scientific Notation

Write the following numbers in standard form, adjusting the mantissa so that it is 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 meters using standard form.

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

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

Round the following numbers so that they have exactly three significant figures and then express them in standard form:

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

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

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

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

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

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

Work out the following multiplications and divisions mentally (without using a calculator), applying the properties of powers, 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) $(1,5 \cdot 10^6) \cdot (4 \cdot 10^{-2})$

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 format.

Work out the following additions and subtractions of numbers in standard form that have the same exponent. Take out the 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) $5,15 \cdot 10^{12} + 2,03 \cdot 10^{12}$

Work out the following additions and subtractions by first adjusting the exponents to be able to take 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) $9,2 \cdot 10^{-5} - 6,1 \cdot 10^{-6}$

Work out the following combined operation and express the final result in standard form, rounding to three significant figures:

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

A server hard drive has a storage capacity of $2$ TB (Terabytes). Given that $1\text{ TB} = 1024\text{ GB}$, $1\text{ GB} = 1024\text{ MB}$, $1\text{ MB} = 1024\text{ KB}$ and $1\text{ KB} = 1024\text{ B}$ (Bytes). Calculate exactly how many bytes the hard drive can store and express the result in standard form.

Given the physical quantities $A = 3,2 \cdot 10^5$ and $B = 5,1 \cdot 10^{-4}$, calculate the value of the algebraic expression $\frac{A^3}{\sqrt{B}}$. Express the final result in standard form with three significant figures and clearly state its order of magnitude.

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

The total volume of water in all the Earth's oceans is estimated at $1,332 \cdot 10^9\text{ km}^3$. If we assume that a standard drop of water has an average volume of $0,05$ ml, determine the total number of drops of water in the oceans. Express the result in standard form. (Remember that $1\text{ ml} = 1\text{ cm}^3$).

When measuring the distance between two cities using a GPS system, a real value of $456321$ m is obtained. A tourist map approximates this distance giving a value of $4,56 \cdot 10^5$ m. Calculate the absolute error and the relative error made when using the map's approximation. Express both errors in standard form.

The edge of a perfect cube has been measured with a precision instrument and its length is known to be in the interval $[1,25 \cdot 10^{-2}, 1,26 \cdot 10^{-2}]$ m. Determine the interval in which the volume of the cube lies, expressed in $\text{mm}^3$, giving the endpoints in standard form. Calculate the absolute error bound if we take the midpoint of this interval as the estimated volume.

A maximum security encryption algorithm uses $256$-bit keys, which means that the total number of possible keys is $2^{256}$. Using the properties of common logarithms and given that $\log_{10} 2 \approx 0,30103$, express the total number of combinations in standard form $N \cdot 10^n$, where $1 \le N < 10$.

The energy released by an earthquake is empirically related to its magnitude $M$ on the Richter scale by the formula $\log_{10} E = 11,8 + 1,5M$, where $E$ is measured in ergs. Calculate the energy released by an earthquake of magnitude $7,5$ and by another of magnitude $8,5$. How many times greater is the energy of the second compared to the first? Express the energy results in standard form.

A sample of radioactive material used in fossil dating initially contains $N_0 = 4,5 \cdot 10^{20}$ nuclei. The number of nuclei decays exponentially according to the law $N(t) = N_0 \cdot e^{-\lambda t}$, where $\lambda = 3,2 \cdot 10^{-4}\text{ years}^{-1}$. Calculate the number of nuclei that will remain undecayed after $5000$ years. Express the final result in standard form with two significant figures.

Newton's law of universal gravitation states that the 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 centers is $d = 3,84 \cdot 10^5$ km. Calculate the force of attraction $F$ in Newtons, using standard form in all intermediate calculations.

The global Gross Domestic Product (GDP) in a given year was approximately $8,5 \cdot 10^{13}$ dollars. If the world population at that time was $7,8 \cdot 10^9$ inhabitants, calculate the average GDP per capita. If the population is projected to grow by $1,2\%$ annually and the GDP to grow by $2,5\%$ annually, estimate the GDP per capita after $10$ years. Express all results in standard form.