# 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!

what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
Jeffrey
Jeffrey
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function