Solve?
5 sqrt 2^3x * 64^2x+2 =32768^x
it might help to write everything in terms of powers of sqrt(2)
64=2^6=sqrt(2)^12
32768=2^15=sqrt(2)^30
So, if u=sqrt(2), we have
5u^(3x) * u^(12(2x+2)) = u^30x
5u^(3x) * u^(24x+24) = u^30x
You sure that 5 belongs there, or is this just problem 5?
Without the 5, x=8 since we then just have to add the powers to get
3x+24x+24=30x
x=8
With the 5, things get messier.
Yeah I'm sure 5 is there because it's power cube of 5sqrt
To solve the equation:
5 * sqrt(2^(3x)) * 64^(2x+2) = 32768^x
Let's break it down step by step:
1. Start by simplifying the terms on both sides of the equation.
Since 2^3x can be written as (2^3)^x = 8^x, and 64 can be written as 8^2, we can simplify as follows:
5 * sqrt(8^x) * (8^2)^(2x+2) = 32768^x
2. Squaring and simplifying further:
5 * sqrt(8^x) * (64)^(2x+2) = (8^5)^x
Squaring the terms on both sides gives:
5^2 * (8^x) * (64)^(2x+2) = (8^5)^x
25 * 8^x * (64^(2x)) * (64^2) = 8^(5x)
3. Now, let's focus on the exponents. Applying the power rule of exponents, we have:
25 * 8^x * (8^(2x))^2 * 8^4 = 8^(5x)
25 * 8^x * 8^(4x) * 8^4 = 8^(5x)
25 * (8^x) * (8^4) * (8^4x) = 8^(5x)
25 * (8^x) * (8^4) * (8^4)^(x) = 8^(5x)
25 * (8^x) * (8^4) * (8^4)^x = 8^(5x)
4. Now, we can set the exponents equal to each other, since the bases are the same:
25 * (8^x) * (8^4) * (8^4)^x = 8^(5x)
25 * (8^x) * (8^4) * (8^4*x) = 8^(5x)
25 * (8^x) * (8^4) * (8^(4x)) = 8^(5x)
25 * (8^x) * (8^4) * (8^(4x)) = 8^(5x)
5. Since all the bases are 8, we can cancel them out:
25 * (8^4) * (8^(4x)) = 8^(5x - x)
25 * (8^4) * (8^(4x)) = 8^(4x)
6. Simplifying further, we get:
25 * 8^(4 + 4x) = 8^(4x)
25 * 8^(4 + 4x) = 8^(4x)
7. Now, we can focus on the exponents again. Applying the power rule of exponents:
25 * 8^(4) * 8^(4x) = 8^(4x)
25 * 8^4 * 8^(4x) = 8^(4x)
200 * 8^(4x) = 8^(4x)
8. Dividing both sides by 8^(4x):
200 = 1
Thus, there is no valid value that solves the equation.