math question powers

[adam2016]

Hi guys,

just wondering if anybody could give me some help with this expression,

(4x^2)^2

the answer is 16x^4

I understand how we got ^4 because we multiplied the exponents but how did we get 16?

do you need to multiply 4x by 2 aswell?

thanks

[helios]

Firstly, 4x^2 means (x^2)*4, not (x*4)^2

Secondly, exponentiation distributes over multiplication:
(x*y)^z = x^z * y^z

(4 * x^2)^2 = 4^2 * (x^2)^2 = 16 * (x^2)^2 = 16 * x^(2*2) = 16 * x^4

[adam2016]

thanks helios =)

[adam2016]

hey Helios just a second question on the same subject

in the example on this youtube video
https://www.youtube.com/watch?v=rEtuPhl6930

(2^10)^5

with (4x^2)^2 = 4x^2 * 4x^2 = 16x4 right?

but with (2^10)^5

for some reason in the example the base doesn't change
so it is 2^10 * 2^10 * 2^10 * 2^10 * 2^10 = 2^50

but why doesn't the base change here shouldn't it not be 2 * 2 * 2 * 2 * 2 = 32^50?

thanks

[adam2016]

also how did you get 16?

(4 * x^2)^2 = 4^2 * (x^2)^2 = 16 * (x^2)^2 = 16 * x^(2*2) = 16 * x^4

[helios]

but with (2^10)^5

for some reason in the example the base doesn't change

Exponentiating an exponentiation doesn't change the base of the inner exponentiation. (x^y)^z = x^(y*z)
Again, ax^2 means a(x^2), not (ax)^2, therefore (ax^2)^2 = a^2 x^4. You need to drill this bit of notation into your head, otherwise nothing is ever going to make sense.

also how did you get 16?

I specifically only changed one term at a time to make it as clear as possible what was happening. Look closely at the transformation.
4^2 * (x^2)^2 = 16 * (x^2)^2

[mbozzi]

Remember that
a^2 = a * a
So
(a^2)^2 = a^2 * a^2 = a * a * a * a = a^4
And equivalently
ca^2 = c * a * a
So
(ca^2)^2 = c*a*a * c*a*a = c^2a^4;