Question

Simplify.

Answer 1

\[(4xy ^{2})^{3}(xy)^{5}\]

Answer 2

@phi

Answer 3

look at just the \( (xy)^5 \)
remember what ^5 means: multiply by itself 5 times
xy * xy * xy * xy * xy

xy is short for x*y so we could write that as
x*y*x*y*x*y*x*y*x*y

the order of multiplying can be changed around, so that is also the same as
x*x*x*x*x*y*y*y*y*y

and we can use the short-hand idea to rewrite that as
\[ x^5 y^5\]

Answer 4

Would I have to add the other exponents along with it?

Answer 5

there are short cuts to figuring this out, but the long way always works. (and if you learn the long way, it will make you want to learn the short way)

what is
\[ (4xy^2)^3\]

Answer 6

Well wouldn't the be x^4 y^5 ?
Sorry still learning these so.

Answer 7

think of \( (4xy^2) \) as one "thing"
when we put a little 3 in the upper right:
\( (4xy^2)^3 \) that means multiply by itself 3 times.
can you write that down, showing \( (4xy^2) \) times itself 3 times.

Answer 8

4xy^6

Answer 9

I think that's what it is...

Answer 10

I guess it is not clear that x^3 means x*x*x
and (7xy)^3 means (7xy)*(7xy)*(7xy)
and (abcd)^3 means (abcd)(abcd)(abcd)
try again

Answer 11

Oh so the 4 would also go to the third power?

Answer 12

4xy^2 * 4xy^2 * 4xy^2, correct?

Answer 13

yes

Answer 14

now if you expand y^2 to y*y
then
4xy^2 * 4xy^2 * 4xy^2 becomes
4*x*y*y*4*x*y*y*4*x*y*y

if you re-arrange that you get
4*4*4 * x*x*x * y*y*y*y*y*y

Answer 15

which can be written using the short-cut as
4^3 * x^3 * y^6

so what we have so far is
\[ (4xy ^{2})^{3}(xy)^{5} \\ 4^3x^3y^6x^5y^5
\]

Answer 16

Then it would be \[64x ^{8}y ^{11}\]

Answer 17

exactomundo!

Answer 18

to do this the fast way:
\[\left( x^a \right)^b = x^{ab} \]
Example:
\[ (x^3)^2 = x^6 \]
or
\[ (x^1 y^2)^4 = x^{1\cdot 4} y^{2\cdot 4} = x^4y^8\]

but if that does not make sense, you can go back to basics:
\[ (x^1 y^2)^4 = (xyy)(xyy)(xyy)(xyy) = (xxxxyyyyyyyy)= x^4y^8 \]

Answer 19

Actually that is how I did it xD Thank you so much for your help c: