Math Help? Easiest way to slove this. (7x)^14 = (14x)^7?

Question

(7x)^14 = (14x)^7

sloving for x.

Answer 1

(7x)^14=(14x)^7
7^14 * x^14=14^7 * x^7
one solution willbe x=0
x^7=14^7/7^14
x^7=(14/49)^7
you will get 7 roots for this, pricipal root will be x=14/49=2/7

Answer 2

(7x)^14 = (14x)^7

First things first; we apply the exponential property
(ab)^(m) = (a^m)(b^m)

(7^14)x^14 = (14^7)x^7

Now, move everything to the left hand side.

(7^14)x^14 - (14^7)x^7 = 0

Note that 14 = 2*7, so using the same exponential property as earlier,

(7^14)x^14 - (2^7)(7^7)x^7 = 0

At this point, it is clear that (7^7)x^7 is a common factor, if not the greatest common factor. Let's factor that out,

(7^7)x^7 [ (7^7)x^7 - (2^7) ] = 0

Equating each of these to 0:

(7^7)x^7 = 0, (7^7)x^7 - (2^7) = 0

Let's solve these individually:

(7^7)x^7 = 0
x^7 = 0, which implies
x = 0. {Note: I am purposely excluding the complex solutions, since there would be 6 of them here.}

(7^7)x^7 - (2^7) = 0

(7^7)x^7 = 2^7

Taking the 7th root of both sides, we get

7x = 2, which means x = 2/7

Therefore, the distinct real solutions are x = {0, 2/7}

Answer 3

Easiest is to expand

(7x)^14 = (14x)^7

7^14 * x^14 = 14^7 * x^7

x^15 / x^7 = 14^7 / 7^14

x^7 = (2*7)^7/7^14 = 2^7 * 7^7 / 7^14

x^7 = 2^7/7^7

x = 7th root of (2^7/7^7)
= 2/7

Answer 4

First take the 7th root of both to get:
(7x)^2 = 14x
so 49x^2 = 14x

Divide both sides by 7 and factor to get:
7x(7x - 2) = 0

So x = 0, 2/7

Answer 5

(7x)^14 = (14x)^7
(7^14)(x^14)=(14^7)(x^7) divide by 7^7 you get
(7^7)(x^14)=(2^7)(x^7) divide by x^7 you get
(7^7)(x^7)=(2^7) divide by 7^7 you get
x^7 = (2/7)^7
x = 2/7
also x = 0 in the original equation this root was lost when I divided both sides by x^7
So two answers x=0 and x= 2/7