Given that (2^3x) x (7^2x) = 7^(x-1) x 2^(2x+2), find 14^x without using a calculator.
To find the value of 14^x without using a calculator, we can simplify the given equation and make it in terms of 14^x.
Let's start by simplifying both sides of the equation.
On the left-hand side, we have (2^3x) x (7^2x).
Now, let's simplify this expression using the properties of exponents:
2^3x can be written as (2^3)^x which equals 8^x, and similarly, 7^2x can be written as (7^2)^x which equals 49^x.
So, the left-hand side can be further simplified as 8^x x 49^x.
On the right-hand side, we have 7^(x-1) x 2^(2x+2).
Let's simplify this expression as well:
7^(x-1) can be rewritten as (7^1)^x-1 which equals 7^x-1, and 2^(2x+2) can be rewritten as (2^2)^x x 2^2 which equals 4^x x 4.
So, the right-hand side can be further simplified as 7^x-1 x 4^x x 4.
Now, our equation becomes:
8^x x 49^x = 7^x-1 x 4^x x 4.
To simplify further, let's rewrite both sides using a common base, such as 2.
8 can be written as 2^3 and 49 can be written as 7^2.
So, our equation becomes:
(2^3)^x x (7^2)^x = 7^x-1 x 4^x x 4.
Simplifying each side gives us:
2^(3x) x 7^(2x) = 7^(x-1) x 4^x x 4.
Now, let's arrange the terms so that the bases are the same on both sides.
Re-arranging the right-hand side gives us:
7^(x-1) x 4^x x 4 = 7^(2x) x 2^2.
Since the bases are the same, we can equate the exponents of the bases.
Equating the exponent of 7 on both sides gives us:
x - 1 = 2x.
Simplifying the equation results in:
-x = -1.
Now, let's solve for x:
Dividing both sides by -1 gives us:
x = 1.
So, we have found the value of x to be 1.
Finally, substituting this value of x in 14^x gives us:
14^x = 14^1 = 14.
Therefore, 14^x is equal to 14.