Exercises: Standard form

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) $456 \cdot 10^{-4}$

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

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

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) $5,15 \cdot 10^{12} + 2,03 \cdot 10^{12}$

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) $98,99$

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) $-45,2 \cdot 10^{-8}$

In an exam, Sergio must calculate the following sum and give the result in scientific notation:

Sergio writes on his paper:

"Since $3,5 + 2,5 = 6,0$ and $10^4 \cdot 10^5 = 10^9$, the result is $6,0 \cdot 10^9$".

1. a) Identify the two conceptual errors made by Sergio when performing the operation and explain why his reasoning contradicts the properties of powers and factoring out a common factor.
2. b) Solve the operation correctly step by step, showing the prior equalization of exponents, and express the result in formal scientific notation.

According to data from the Institute for the Diversification and Saving of Energy (IDAE), the average electricity consumption per household in Spain is about $9\text{ kWh}$ per day, equivalent to $3,24 \cdot 10^7\text{ J}$. We want to calculate the total annual energy consumed by a residential development of $50$ dwellings ($5 \cdot 10^1$ households) during a standard year of $365$ days ($3,65 \cdot 10^2$ days).

1. a) Without using a calculator, make a prior estimation by rounding the significands to the nearest integer to predict exclusively the order of magnitude (the power of $10$) of the final result.
2. b) Perform the exact algebraic calculation, expressing the result in scientific notation.
3. c) Check whether your initial estimation matched the order of magnitude obtained, and argue why estimating the exponent beforehand helps avoid typical errors when entering powers into a calculator.

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:

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

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.

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.

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 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.

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

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

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

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

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

In a microelectronics laboratory, two particles are analyzed whose electric charges, $A$ and $B$, are given in strict scientific notation: $A = a \cdot 10^p$ and $B = b \cdot 10^q$, with $1 \le a, b < 10$ and $p, q \in \mathbb{Z}$. Their individual values are unknown, but their product and quotient have been determined experimentally:

1. a) Set up the system relating both quantities and solve algebraically for the exact value of $A$ and $B$.
2. b) Write $A$ and $B$ in formal scientific notation, clearly specifying the significand and order of magnitude for each, and verify that the significands obtained strictly satisfy the condition $1 \le |m| < 10$.

According to reports from the Ministry for the Ecological Transition and the Demographic Challenge (MITECO), the total maximum capacity of reservoirs in Spain is $5,60 \cdot 10^4\text{ hm}^3$. After a year of scarce rainfall, the peninsular water reserve is at $38,0\,\%$ of that maximum capacity.

Data: $1\text{ hm}^3 = 10^6\text{ m}^3 = 10^9\text{ L}$.

1. a) Model the situation by expressing the volume of stored water in scientific notation, both in cubic meters ($\text{m}^3$) and in liters ($\text{L}$).
2. b) If the estimated combined consumption for all uses (urban, irrigation, and industrial) is $3,20 \cdot 10^7\text{ m}^3$ per day and all new water inflow were completely cut off, calculate how many full days of water supply would remain. Express the division in scientific notation and translate the final result into whole days, stating its order of magnitude.

The total volume of water in all Earth's oceans is estimated at $1,332 \cdot 10^9\text{ km}^3$. 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 $1\text{ ml} = 1\text{ cm}^3$).

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 to three significant figures and clearly state its order of magnitude.

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.

Global Gross Domestic Product (GDP) in a given year was approximately $8,5 \cdot 10^{13}$ euros. If the global population at that time was $7,8 \cdot 10^9$ 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 standard form.

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 $N \cdot 10^n$, where $1 \le N < 10$.

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

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 undecayed nuclei remaining after $5000$ years. Express the final result in standard form to two significant figures.

To carry out the addition of astronomical quantities:

a group of students proposes three working methods:

• Method 1: Shift all the decimal points to equalize the three terms to the lowest common exponent ($10^8$), operate with the resulting significands, and normalize to scientific notation at the end.

• Method 2: Shift the decimal points to equalize all terms to the order of magnitude of the dominant term ($10^{11}$), operate with the decimal significands, and apply the criterion of significant figures.

• Method 3: Expand all quantities by writing out all their zeros in standard decimal notation, perform column arithmetic, and convert the resulting answer to scientific notation.

1. a) Apply Methods 1 and 2 in detail, showing the intermediate numerical transformations.
2. b) Comparatively evaluate the three approaches. Justify why Method 2 is the most appropriate and consistent with experimental work in applied sciences, analyzing what happens to the significant figures of the term $2,50 \cdot 10^8\text{ m}$ compared to the precision of the primary measurement, and explain why Methods 1 and 3 are inefficient or physically meaningless.

Consider two strictly positive real numbers expressed in scientific notation: $X = a \cdot 10^p$ and $Y = b \cdot 10^q$, with $p, q \in \mathbb{Z}$ and satisfying the canonical condition $1 \le a < 10$ and $1 \le b < 10$.

1. a) Formally prove that if $p > q$, then necessarily $X > Y$, relying rigorously on the inequalities bounding the significands $a$ and $b$, without using particular numerical examples.
2. b) Irene states: "When adding two numbers in scientific notation $X + Y$ with $p > q$, the order of magnitude of the result is always exactly $p$". Analyze whether Irene's statement is true or false. If it is false, provide a numerical counterexample in scientific notation and deduce the exact analytical condition that the sum of the aligned significands must satisfy for the order of magnitude to become $p + 1$. Furthermore, prove whether it is mathematically possible or impossible for the order of magnitude of the sum to reach $p + 2$.