Question

Factor this expression completely
9a4 - b2

Answer 1

That's a difference of two squares; that's the result you get when you multiply something like
\[(x-y)(x+y) = x^2 - xy + xy - y^2 = x^2 - y^2\]
Can you figure out what that means for you in this problem?

Answer 2

no will u explain it please sorry

Answer 3

Okay, if we match up your equation
\[9a^4 - b^2\]with mine
\[x^2-y^2\]then
\[x^2=9a^4\]and\[y^2 = b^2\]right?

Answer 4

yes but I don't understand this at all

Answer 5

im so sorry

Answer 6

okay, can you solve my two little equations for x and y?

Answer 7

sure wat r they

Answer 8

\[x^2 = 9a^4\]
\[y^2 = b^2\]

Answer 9

okay that's my problem but I don't know how to solve it

Answer 10

Well, all the second one says is that there is some expression, \(y\), that if you multiply it by itself, it gives you \(b^2\). What would that be?

Answer 11

Isn't that just \(b\)? \(b*b = b^2\) right?

Answer 12

o ya okay I get that

Answer 13

Well, how about \[x^2 = 9a^4\]\(x\) is some expression that if you multiply \(x*x\) you get \(9a^4\). Any ideas? It will have a number, and it will have a letter...

Answer 14

(x*x=x^2)
(9a^4= )
im confused now

Answer 15

okay, what is the square root of 9?

Answer 16

I think 3 right @whpalmer4

Answer 17

does 3*3= 9? if so, then the answer is "yes, I'm sure of it!" :-)

Answer 18

okay how else would I do this

Answer 19

Okay, so we're going to have 3<something>*3<something> = 9a^4
Any guesses about the <something>?

Answer 20

3*3=9a^4

Answer 21

3<something>*3<something> = 9<something>^2

Answer 22

u confued me

Answer 23

Do you remember the properties of exponents?

Answer 24

nope

Answer 25

9*9=9^2

Answer 26

One of them is \[a^n*a^m = a^{n+m}\]
Or if the two exponents are equal, that becomes
\[a^n*a^n = a^{2n}\]

Answer 27

So we want to multiply two equal things together and get \(a^4\)...what does that property I just showed you suggest we might use?

Answer 28

distropit idk my to spell it

Answer 29

no, I'm looking for a value...

Answer 30

idk
im confused

Answer 31

Okay, \[a^n*a^n = a^{2n}\]We are trying to find something that when multiplied by itself gives you \(a^4\). Isn't \(a^4\) the same as \(a^{2n}\) if \(n=2\)?

Answer 32

idk the letters r confusing me

Answer 33

does 4 = 2n, if n = 2?

Answer 34

can u draw it out

Answer 35

Does 4 = 2*2?

Answer 36

yes 2*2=4

Answer 37

Okay, so does 4 = 2n, if n = 2?

Answer 38

yes

Answer 39

okay we does that have to do with my math

Answer 40

Excellent! :-)

So that means that \[a^2*a^2 = a^4\]
And we can package that up with \[3*3 = 9\]and we get
\[3a^2*3a^2 = 9a^4\]
So we can factor our expression as
\[(3a^2-b)(3a^2+b) = 9a^4-b^2\]

Answer 41

Because\[(3a^2-b)(3a^2+b) = 9a^4 + 3a^2b-3a^2b-b^2 = 9a^4 -b^2\]