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

Jul 26, 2015

$x = 2$

#### Explanation:

Here's how you can quickly solve this equation.

${4}^{x} \cdot {3}^{2 x - x} = 144$

Use the power of a power property of exponents to write

${4}^{x} = {\left({2}^{2}\right)}^{x} = {2}^{2 x}$

The equation becomes

${2}^{2 x} \cdot {3}^{x} = 144$

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

${2}^{2 x} \cdot {3}^{x} = {2}^{4} \cdot {3}^{2}$

This is equivalent to having

${2}^{2 x} = {2}^{4}$ and ${3}^{x} = {3}^{2}$, which in turn will get you

$x = \textcolor{g r e e n}{2}$, which is verified by

$2 x = 4 \implies x = \frac{4}{2} = \textcolor{g r e e n}{2}$,

Jul 26, 2015

Simplify the equation to ${12}^{x} = {12}^{2}$, hence $x = 2$

#### Explanation:

${4}^{x} \cdot {3}^{2 x - x} = {4}^{x} \cdot {3}^{x} = {\left(4 \cdot 3\right)}^{x} = {12}^{x}$

$144 = {12}^{2}$

So the original equation simplifies to: ${12}^{x} = {12}^{2}$

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