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

2 Answers
Jul 26, 2015

Answer:

#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)#,

Jul 26, 2015

Answer:

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#