# 1.2 Exponents and scientific notation  (Page 6/9)

 Page 6 / 9

Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.

$4.7\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-13}$

## Scientific notation

A number is written in scientific notation    if it is written in the form $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{n},$ where $\text{\hspace{0.17em}}1\le |a|<10\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is an integer.

## Converting standard notation to scientific notation

Write each number in scientific notation.

1. Distance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 m
2. Diameter of Andromeda Galaxy: 1,300,000,000,000,000,000,000 m
3. Number of stars in Andromeda Galaxy: 1,000,000,000,000
4. Diameter of electron: 0.00000000000094 m
5. Probability of being struck by lightning in any single year: 0.00000143

Write each number in scientific notation.

1. U.S. national debt per taxpayer (April 2014): $152,000 2. World population (April 2014): 7,158,000,000 3. World gross national income (April 2014):$85,500,000,000,000
4. Time for light to travel 1 m: 0.00000000334 s
5. Probability of winning lottery (match 6 of 49 possible numbers): 0.0000000715
1. $1.52×{10}^{5}$
2. $7.158×{10}^{9}$
3. $8.55×{10}^{13}$
4. $3.34×{10}^{-9}$
5. $7.15×{10}^{-8}$

## Converting from scientific to standard notation

To convert a number in scientific notation    to standard notation, simply reverse the process. Move the decimal $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ places to the right if $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is positive or $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ places to the left if $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is negative and add zeros as needed. Remember, if $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is positive, the value of the number is greater than 1, and if $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is negative, the value of the number is less than one.

## Converting scientific notation to standard notation

Convert each number in scientific notation to standard notation.

1. $3.547\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{14}$
2. $-2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{6}$
3. $7.91\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-7}$
4. $-8.05\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-12}$

Convert each number in scientific notation to standard notation.

1. $7.03\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{5}$
2. $-8.16\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{11}$
3. $-3.9\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-13}$
4. $8\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-6}$
1. $703,000$
2. $-816,000,000,000$
3. $-0.000\text{\hspace{0.17em}}000\text{\hspace{0.17em}}000\text{\hspace{0.17em}}000\text{\hspace{0.17em}}39$
4. $0.000008$

## Using scientific notation in applications

Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of water contains around $\text{\hspace{0.17em}}1.32\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{21}\text{\hspace{0.17em}}$ molecules of water and 1 L of water holds about $\text{\hspace{0.17em}}1.22\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{4}\text{\hspace{0.17em}}$ average drops. Therefore, there are approximately $\text{\hspace{0.17em}}3\cdot \left(1.32\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{21}\right)\cdot \left(1.22\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{4}\right)\approx 4.83\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{25}\text{\hspace{0.17em}}$ atoms in 1 L of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!

Need help solving this problem (2/7)^-2
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_