Question

What value of n solves the equation?

4^n = 1/64

A.
–3

B.
3

C.
16

D.
256

Answer 1

please note that:
\[4^{n}=(2^{2})^{n}=2^{2n}\]
and
\[\frac{ 1 }{ 64 }=64^{-1}=(2^{6})^{-1}=2^{-6}\]
so please substitute the above equalities into your equation

Answer 2

I still dont get it, we just started this in class today but im still really confused by all of it.

Answer 3

please do you understand the equalities that I wrote above?

Answer 4

Not really i asked my sister if she could help me but she said she doesnt understand it either :/ im not the best at math.

Answer 5

ok! Im going to explaint them you now!

Answer 6

Thank you!

Answer 7

we have \[\frac{ 1 }{ 64 }\]
now in that fraction the exponent of 64 is 1, do you agree?

Answer 8

But wouldnt i be -1? My teacher said when you have a negative exponent it turns into a fraction.

Answer 9

it*

Answer 10

I know what you say, nevertheless we have :
\[(64)^{1}=64\]
do you agree?

Answer 11

Yes

Answer 12

ok! now there is a property of powers which is:
\[\frac{ 1 }{ (a)^{m} }=a ^{-m}\]
please be careful, that above is a property
do you agree?

Answer 13

This is the part that im confused on, but i think i understand. I cant remember what the (a) is. But i think its a property

Answer 14

in our casgeneral e acan be any real positive number, whereas n can be any unteger positive number. Now in our case a=64 and m=1

Answer 15

sorry: in general case a can be any real positive number....

Answer 16

ohh okay that would mean 1/a^1 correct?

Answer 17

\[\frac{ 1 }{ a ^{1} }\]
yes it is correct

Answer 18

Okay i understand that now

Answer 19

so if we apply our property, we can write:
\[\frac{ 1 }{ 64 }=(64)^{-1}\]
do you agree?

Answer 20

yes

Answer 21

now, please note that:
\[64=4*4*4=2^{2}*2^{2}*2^{2}=2^{6}\]
because, ise multiplication of powers with equal base is a power whose base is the same base and the exponent is the sum of the exponents of the powers which we want to multiply

Answer 22

-3. Due to multiplication tables.

Answer 23

Okay i understand that part