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) $0.0023 \times 10^5$
4. d) $456 \times 10^{-4}$

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) $12.4 \times 10^3$
2. b) $0.85 \times 10^{-2}$
3. c) $1 \times 10^6$
4. d) $-45.2 \times 10^{-8}$

The diameter of a human red blood cell is approximately $0.0075$ mm. Express this quantity in meters using scientific notation.

The estimated wealth of one of the richest people in the tech 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) $4.05 \times 10^8$ m
2. b) $1.200 \times 10^{-5}$ kg
3. c) $9 \times 10^{12}$ s

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

1. a) $4567892$
2. b) $0.00034091$
3. c) $98.99$

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

$A = 0.0045 \times 10^{-3}$

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

$C = 0.45 \times 10^{-5}$

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

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) $(3 \times 10^4) \times (2 \times 10^5)$
2. b) $\frac{8 \times 10^{-3}}{4 \times 10^{-7}}$
3. c) $(1.5 \times 10^6) \times (4 \times 10^{-2})$

The mass of the Earth is approximately $5.97 \times 10^{24}$ kg and that of the planet Jupiter is $1.90 \times 10^{27}$ 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 $3 \times 10^5$ km/s. A "light-year" is the distance light travels in a vacuum in one Earth year ($365.25$ days). Calculate how many kilometers a light-year corresponds to, and express the result in scientific notation to four significant figures.

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) $3.2 \times 10^5 + 4.5 \times 10^5$
2. b) $8.7 \times 10^{-4} - 3.2 \times 10^{-4}$
3. c) $5.15 \times 10^{12} + 2.03 \times 10^{12}$

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) $4.5 \times 10^7 + 3.8 \times 10^8$
2. b) $9.2 \times 10^{-5} - 6.1 \times 10^{-6}$

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

$\frac{(2.5 \times 10^6) \times (4.2 \times 10^{-3})}{3.1 \times 10^{-4} + 1.9 \times 10^{-5}}$

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 the exact number of bytes the hard drive can store and express the result in standard form.

Given the physical quantities $A = 3.2 \times 10^5$ and $B = 5.1 \times 10^{-4}$, calculate the value of the algebraic expression $\frac{A^3}{\sqrt{B}}$. Express the final result in standard form to three significant figures and clearly state its order of magnitude.

Global Gross Domestic Product (GDP) in a given year was approximately $170 \times 10^{13}$ pesos. If the global population at that time was $7.8 \times 10^9$ inhabitants, calculate the mean GDP per capita. If the population is projected to grow by $1.2\%$ per year and GDP is projected to grow by $2.5\%$ per year, estimate the GDP per capita after $10$ years. Express all results in scientific notation.

A high-security encryption algorithm uses $256$-bit keys, which means the total number of possible keys is $2^{256}$. Using the properties of common logarithms (base 10) and given that $\log_{10} 2 \approx 0.30103$, express the total number of combinations in standard form $N \times 10^n$, where $1 \le N < 10$.

A sample of radioactive material used in fossil dating initially contains $N_0 = 4.5 \times 10^{20}$ nuclei. The number of nuclei decays exponentially according to the law $N(t) = N_0 \times e^{-\lambda t}$, where $\lambda = 3.2 \times 10^{-4}\text{ years}^{-1}$. Calculate the number of undecayed nuclei remaining after $5000$ years. Express the final result in standard form to two significant figures.

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.

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 $4.56 \times 10^5$ 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 lie in the interval $[1.25 \times 10^{-2}, 1.26 \times 10^{-2}]$ m. Determine the interval in which the volume of the cube lies, expressed in $\text{mm}^3$, giving the end points in standard form. Calculate the absolute error bound if we take the midpoint of this interval as the estimated volume.

Given that $18$ grams of water contain $6.022 \times 10^{23}$ 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.

The total volume of water in all Earth's oceans is estimated at $1.332 \times 10^9\text{ km}^3$. Assuming 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 $1\text{ ml} = 1\text{ cm}^3$).

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 $G = 6.67 \times 10^{-11}\text{ N}\times\text{m}^2/\text{kg}^2$, the mass of the Earth is $m_1 = 5.97 \times 10^{24}$ kg, the mass of the Moon is $m_2 = 7.35 \times 10^{22}$ kg and the mean distance between their centers is $d = 3.84 \times 10^5$ km, calculate the force of attraction $F$ in newtons, using standard form in all intermediate calculations.