Exercises: Standard form and significant figures

Light travels at $300\,000\,000\ \text{m/s}$. Write this speed in standard form.

A red blood cell has a diameter of about $0.000008\ \text{m}$. Express this length in standard form.

Write the number $4.5\times10^{3}$ using all its digits.

An aircraft carrier has a mass of about $9.7\times10^{7}\ \text{kg}$. Write this mass using all its digits.

Of these three ways of writing the same number, only one is in correct standard form. State which one it is and explain what is wrong with the other two:

1. a) $25\times10^{4}$
2. b) $2.5\times10^{5}$
3. c) $0.25\times10^{6}$

Fill in the blanks: $6\,000 = 6\times10^{\dots}$ and $0.003 = 3\times10^{\dots}$.

One day has $86\,400$ seconds. Write it in standard form.

Which is larger: $2\times10^{5}$ or $3\times10^{4}$? Write them using all their digits and compare them.

Order these numbers written in standard form from smallest to largest: $3.2\times10^{4}$; $9.8\times10^{3}$; $1.5\times10^{5}$; $4\times10^{4}$. Explain what you look at first to compare them: the leading digits or the exponents.

The mass of an ant is about $3\times10^{-3}\ \text{g}$ and that of an elephant is about $6\times10^{3}\ \text{kg}$. a) Express the mass of the elephant in grams, in standard form. b) With both masses in grams, which has the larger exponent? Which of the two animals has a greater mass?

How many times larger is $6\times10^{8}$ than $2\times10^{5}$? Do this by dividing the powers of ten and give the result in standard form.

How many significant figures does each of these measurements have? Explain the criterion you followed in each case.

1. a) $2.50\ \text{cm}$
2. b) $0.034\ \text{kg}$
3. c) $1.02\ \text{L}$

Vera's calculator displays $7.8462$ as the result of a measurement. Round it to two significant figures and then to three.

Using the ruler in your pencil case, you measure the thickness of a coin and get $0.2\ \text{cm}$. a) How many significant figures does this measurement have? b) Why does it not make sense to write $0.2000\ \text{cm}$ if the ruler only marks millimeters?

Calculate $(3\times10^{4})\times(2\times10^{3})$ and express the result in standard form.

The population of Spain is about $47\,615\,034$ inhabitants. Write it in scientific notation rounded to three significant figures.

Light from the Sun takes about $8\ \text{min}$ and $20\ \text{s}$ to reach the Earth. a) Convert this time into seconds. b) Using the speed of light, $3\times10^{8}\ \text{m/s}$, calculate the distance from the Sun to the Earth in meters, in scientific notation. c) Also express it in kilometers.

A nanometer is a billionth of a meter: $1\ \text{nm} = 10^{-9}\ \text{m}$. Nora's hair is $5\times10^{-5}\ \text{m}$ thick. a) How many nanometers is it? b) How many times thicker is it than a water molecule, which measures about $3\times10^{-10}\ \text{m}$? Give the result rounded to one significant figure.

The teacher writes on the board: "mass of the Earth: $5.97\times10^{24}\ \text{kg}$". a) How many significant figures does this value have? b) Write it rounded to a single significant figure. c) What is lost when rounding like this? Explain in what situations it makes sense to do so.

Ximena calculates the density of a stone and the calculator shows $2.68473\ \text{g/cm}^3$. The balance she used measured to three significant figures, and the measuring cylinder to only two. a) Knowing that a result cannot be more precise than the least precise value, to how many significant figures should Ximena give her result? b) Write it correctly rounded.

Order these three distances from smallest to largest, first converting all of them into meters and scientific notation:

1. a) A large bridge of $4\times10^{6}\ \text{mm}$.
2. b) A cycling stage of $1.8\times10^{2}\ \text{km}$.
3. c) A flight of $9\times10^{5}\ \text{m}$.

How many seconds are in a year? a) Calculate it knowing that a year has $365$ days, a day $24\ \text{h}$, an hour $60\ \text{min}$, and a minute $60\ \text{s}$. b) Express the result in standard form to two significant figures.

A $256\ \text{GB}$ memory card holds about $2.56\times10^{11}$ bytes and each photo on the cell phone takes up about $4\times10^{6}$ bytes. a) How many photos fit on the card? b) Express the result rounded to one significant figure. c) Explain why here it makes sense to give one significant figure and not six.

Sound travels through air at $3.4\times10^{2}\ \text{m/s}$ and light at $3\times10^{8}\ \text{m/s}$. a) How many times faster is light than sound? Round to one significant figure. b) Use this result to explain why in a thunderstorm you see the lightning first and hear the thunder afterwards.