Question

Can someone please help me solve for x!
64^(-x)=4^(-3x)

Answer 1

\[\Large\bf\sf \color{royalblue}{64\quad=\quad 4^3}\]
Let's use this in our expression:\[\Large\bf\sf \left(\color{royalblue}{64}\right)^{-x}\quad=\quad 4^{-3x}\]

Answer 2

\(\bf 64^{-x}=4^{-3x}\qquad {\color{blue}{ 64\implies 2^6\qquad 4\implies 2^2}}
\\ \quad \\
64^{-x}=4^{-3x}\implies (2^6)^{-x}=(2^2)^{-3x}\)

Answer 3

Becomes:\[\Large\bf\sf \left(\color{royalblue}{4^3}\right)^{-x}\quad=\quad 4^{-3x}\]

Answer 4

how do i figure out what x is equal to though?

Answer 5

Using rules of exponents can be written as,
\[\Large\bf\sf 4^{-3x}\quad=\quad 4^{-3x}\]The exponents multiply, remember your rule? :o

Answer 6

no i can't remember it :(

Answer 7

would x be 0?

Answer 8

or would that mean that it is a true statement and i do not solve for x? :O

Answer 9

Mmm Ya I guess 0 would work.
0 is a fairly trivial solution with exponential functions though :[

Answer 10

Since the bases are now equal, it means the exponents have to be equal as well.\[\Large\bf\sf 4^{-3x}=4^{-3x}\qquad\implies\qquad -3x=-3x\]

Answer 11

-3x=-3x, solve for x! :O

Answer 12

well when i do that i get x=1, is that correct? :o

Answer 13

Hmmm x=1 seems to also work for a solution.
Hmmmmm what about x=2? Does that also hold true? :U

Answer 14

yes i believe so :o so even though the question says to solve for x it wouldnt just be 1 or 0?

Answer 15

Mmmm ok I'll stop being so sneaky :) lol
Since the left and right side are completely identical, it means they will share EVERY VALUE OF X.

If you wanted to `solve for x` though,
\[\Large\bf\sf -3x=-3x\]We would add 3x to each side,\[\Large\bf\sf 0=0\]^This expression is true. Zero does in fact equal zero.
So that tells us that the relationship is `true` for all x.

Answer 16

If we had ended up with something like 0=1.
That would tell us that the relationship is `false` for all x.

Answer 17

oh i see! thank you very much! :) can you help me with another one similar to this? its just really hard :(

Answer 18

Sure :U