Solve for x, when (2^(x+1))^3*4^x=8192
To solve for x in the equation (2^(x+1))^3 * 4^x = 8192, we can simplify the equation step by step.
Let's start by simplifying the expression inside the parentheses: 2^(x+1).
Using the exponent rule, where (a^m)^n = a^(m*n), we can rewrite the expression as follows: 2^(x+1) = 2 * 2^x.
Now, substituting this simplified expression back into the original equation, we have (2 * 2^x)^3 * 4^x = 8192.
Next, let's simplify the exponents. By applying the exponent rule again, we get (2^1 * 2^x)^3 * 4^x = 8192.
Simplifying further, we have (2^(1+x))^3 * 4^x = 8192.
Since 4 can be written as 2^2, we can substitute 4^x with (2^2)^x, which simplifies to 2^(2x).
So, the equation becomes (2^(1+x))^3 * 2^(2x) = 8192.
Now, to simplify the equation, we can use the exponent rule once again: (a^m)^n = a^(m*n).
Applying this rule, we have 2^(3(1+x) + 2x) = 8192.
Expanding the exponent expression inside the parentheses, we get 2^(3+3x+2x) = 8192.
Simplifying further, we have 2^(5x+3) = 8192.
Now, we can rewrite 8192 as a power of 2. 8192 is equal to 2^13.
So, we have 2^(5x+3) = 2^13.
To solve for x, we can equate the exponents: 5x + 3 = 13.
Next, we can solve for x by isolating it on one side of the equation: 5x = 13 - 3.
Simplifying, we get 5x = 10.
Finally, dividing both sides of the equation by 5, we find the solution for x: x = 10/5.
Therefore, the solution to the equation (2^(x+1))^3 * 4^x = 8192 is x = 2.