Question

How do you solve `4^x*3^(2x-x) = 144` for `x` ?

Answer 1

`x=2`

Explanation:

Here's how you can quickly solve this equation.

`4^x * 3^(2x-x) = 144`

Use the power of a power property of exponents to write

`4^x = (2^2)^x = 2^(2x)`

The equation becomes

`2^(2x) * 3^(x) = 144`

Now focus on writing the prime factors of `144`

`{:(144 : 2 = 72), (72 : 2 = 36), (36 : 2 = 18), (18 : 2 = 9) :}} -> 2^(4)`

`{:(9 : 3 = 3),(3:3 = 1) :}} -> 3^2`

This means that you have

`2^(2x) * 3^x = 2^4 * 3^2`

This is equivalent to having

`2^(2x) = 2^4` and `3^x = 3^2`, which in turn will get you

`x = color(green)(2)`, which is verified by

`2x = 4 => x = 4/2 = color(green)(2)`,

Answer 2

Simplify the equation to `12^x = 12^2`, hence `x=2`

Explanation:

`4^x*3^(2x-x)=4^x*3^x = (4*3)^x = 12^x`

`144 = 12^2`

So the original equation simplifies to: `12^x = 12^2`

Since exponentiation is a one-one function, the only solution is `x=2`