Question

I am learning about scientific notation, and my book said...

-when multiplying numbers in scientific notation multiply the numbers before the multiplication sign and add the exponent

-for example
(3.0x10^8m/s)x(5.0x10^2s)=15x10^10=1.5x10^11m

However, I was confused why it was 15x10^10 and not 15x100^10 since 10x10=100.

Answer 1

It's because you have a common base of 10, so when you multiply them together youre essentially adding the exponents, just as you would to any other common base, such as x^5 * x^3. It'd be x^8 and you wouldn't account for x*x (or 10*10 in this case). So:

(3 * 10^8m/s) * (5 * 10^2s) = 15*10^10m or you can move divide 10 by 15 and then add 1 to the exponent to account for the division, so 1.5*10^11m.

Hope that clears things up ask me if you're still stuck!

Answer 2

Makes sense! But what's a common base?

@jtug6

Answer 3

It's the bottom number that is raising some power. So like, here are some bases:
5^10, 5 is the base

x^19, x is the base

2.54x10^19, 10 is the base

Answer 4

Why is common base used here but not when we combine like terms?

For eg when I was in my algebra 1 class (if I remember correctly)

x^3*x^3=something around 2x^3 (not sure if three is supposed to be three or 9)

Though, point is, what makes this different from combining them? How do I know when to apply common base?

Answer 5

Uhhh well if the answer was 2x^3 or 2x^9 it'd still be incorrect. x^3 * x^3 is x^6 because you have that x term in common. So when you're multiplying two common bases, you add their exponents as a result. So, x^3 * x^3 is x^(3 + 3) = x^6

Answer 6

x^3 is a short form for x*x*x

Answer 7

ganeshie might have a better job of explaining it, too. ^^ lol

Answer 8

x^3 * x^3 = x*x*x * x*x*x

How many x's do you see on the right hand side ?

Answer 9

6

And ah so this only applies if you have exponents?

Answer 10

Yes, try to see the big picture, its really easy.

Answer 11

For example, if you want to multiply 10 by itself 8 times. We express it as :

10 * 10 * 10 * 10 * 10 * 10 * 10 * 10

Answer 12

That's too painful and waste of time and space

Answer 13

That's one reason we invented the idea of an exponent. That crazy multiplication is expressed in compact form as :

10^8

Answer 14

Do you appreciate how neat `10^8` looks compared to `10 * 10 * 10 * 10 * 10 * 10 * 10 * 10` ?

Answer 15

wow u guys are really good at this math thing, Im trying to do geometry and its really hard

Answer 16

Geometry is indeed hard. But I think you will do great if you know how to use compass and straightedge efficiently...

Answer 17

yes i just started today and i was doing like segment bisectors with them

Answer 18

Heard of euclidea game ?

Answer 19

Yes. It's much easier.

Why do you only keep the base the same after multiplication if two bases are the same?

Answer 20

Let's work the given example and see

Answer 21

You want to simplify
(3.0x10^8m/s)x(5.0x10^2s)

Answer 22

(3.0x10^8)x(5.0x10^2)

is same as

(3.0 x 5.0) x (10^8 x 10^2)

Answer 23

yeah kai plays it, he showed me it

Answer 24

With me so far ?

Answer 25

Its a nice way to learn geometry
http://www.euclidea.xyz/en/game/#/packs/Alpha

Answer 26

yeah, ill check it out, but the tools are too hard to use

Answer 27

Yes

Answer 28

(3.0x10^8)x(5.0x10^2)

is same as

(3.0 x 5.0) x (10^8 x 10^2)

(15) x (10^8 x 10^2)

Answer 29

Let's look at (10^8 x 10^2) closely

Answer 30

10^8 is a short form for 10*10*10*10* 10*10*10*10
10^2 is a short form for 10*10

therefore, 10^8 x 10^2 = 10*10*10*10* 10*10*10*10 x 10*10

How many 10's do you see on the right hand side ?

Answer 31

10

Answer 32

Yeah you will need to get used to the online tools, then only you will enjoy the game :)

Answer 33

10^8 x 10^2 = 10*10*10*10* 10*10*10*10 x 10*10 = 10^10

Answer 34

Notice also that 8+2 = 10.

Can we say 10^8 x 10^2 = 10^(8+2) ?

Answer 35

Yes. Though, I'm curious, why does not adding common bases make sense?

And how does 10^10 turn into 10^11?

Answer 36

We don't need to know what a base is here. We are doing series of multiplications here, thats all.

Answer 37

(3.0x10^8)x(5.0x10^2)

is same as

(3.0 x 5.0) x (10^8 x 10^2)

(15) x (10^8 x 10^2)

(15) x (10^10)

Answer 38

1.5 x 10 = ?

Answer 39

I mean why does it make sense for 10^8x10^2 to = 10^10 and not 100^10

15

Answer 40

Let's do it again from scratch

Answer 41

10^8 = ?

Answer 42

10x10x10x10x10x10x10x10

Answer 43

why, why is it not 100x100x100x100x 100x100x100x100 ?

Answer 44

Because exponents merely tell you how many times the base is going to be multiplied by itself each time

Answer 45

Good. Keep that in mind for the next few minutes.

Answer 46

10^2 = ?

Answer 47

10x10

Answer 48

10^8 = 10x10x10x10x10x10x10x10
10^2 = 10x10
---------------------------------------------
10^8 x 10^2 = ?

Answer 49

multiply the right hand sides

Answer 50

OH I see why it doesn't make sense.

Okay, I understand why we don't multiply the bases.

10^10

So where does 10^11 come from?

Answer 51

All you have to remember is the definition : a^n = a*a*a*... n times

Answer 52

The rules like a^m * a^n = a^(m+n) and everything else follow from that definition

Answer 53

If you remember just the rules and ignore the definition, you will be confused and get stuck everywhere.

Answer 54

(3.0x10^8m/s)x(5.0x10^2s)=15x10^10=1.5x10^11m


Alright that makes sense, but in this problem, 15 turns into 1.5 and 10^10 turns into 10^11? Why?

Answer 55

good night guys

Answer 56

Look at 15x10^10

Answer 57

have good sleep @ILovePuppiesLol

Answer 58

1.5 x 10^1 = ?

Answer 59

15, but why did it change into 1.5?

@ganeshie8

Answer 60

15=1.5*10
150=1.5*10^2
1500=1.5*10^3

Answer 61

Ah, I get it that part now.

@zzr0ck3r

Why did it turn from 1.5*10^11 instead of 1.5*10^10?

Answer 62

What has turned to 1.5*10^11 ?

Answer 63

(3.0x10^8m/s)x(5.0x10^2s)=15x10^10=1.5x10^11m


See how it ends with 10^11

Why?

This is how it was written in my book btw.

Answer 64

15x10^10

first replace 15 by 1.5 x 10^1

Answer 65

what do you get ?

Answer 66

1.5x10^10

Answer 67

You there?

@ganeshie8

Answer 68

sorry kikuo but i not can answering you just in this style ,here in what you need help ?

Answer 69

do you need help in this posted question again ?
i think you will can understanding this multiplication of scientific notations with and other example - how you think ?

Answer 70

an other example

Answer 71

3*10^2 * 5*10^5 =?

Answer 72

or an other one

(2*10^2) * (3*10^8) = ? for your more easy understanding i separated with parentheses

Answer 73

I want to understand why it changed from 10^10 to 10^11

Why did that happen? I want to understand specifically that. : )

Answer 74

@ganeshie8

this 15 *10^10 we transform it in 1,5 *10^11

bc. the scientific notation is in this form just one integer and how many much more the 10 on the necessary power
so than you see 15 is formed from two integer numbers and in this way we can transform it in 1,5 and hence the 10 power will be changed in 11

hope this is understandably for you and helped you

Answer 75

Kikuo do you understand it now ?

Answer 76

Not quite

Answer 77

ask me what no

Answer 78

do you need an other example ?

Answer 79

2*10^3 *5*10^2 = ?

Answer 80

I'm not understanding why the 10 turned into 11.

Like, I don't understand.

Answer 81

\[ 15 \cdot 10^{10}=1.5 \cdot 10^{11} \]
why?

the rule is that we want the number "out front" to be less than 10
15 is bigger than 10, so we have to tweak it.
up above, people showed you
\[ 15 = 1.5 \cdot 10^1 \]
do you see how that is true ?
anyway, if you replace 15 in
\[ 15 \cdot 10^{10}\]
with this new way of writing 15 as 1.5 * 10^1
you get\[ 1.5 \cdot 10^1 \cdot 10^{10} \]
now figure out what \[10^1 \cdot 10^{10} \] is
(you should get \( 10^{11}\))
and the answer is
\[ 1.5 \cdot 10^{11} \]

if this is not clear, please post another question, and people will help some more.