Solve 16(4^2x-1)= 8^x+3
I do not trust your parentheses
eg:
8^(x) + 3 or 8^(x+3)
or
4^(2x) -1 or 4^(2x-1)
if it is
16 [ 4^(2x-1) ] = 8^(x+3)
that is
2^4 [ 2^(4x-2) ] = 2^[3(x+3)]
2^[16x-8] = 2^[3x+9]
or
16-x = 3x+9
16x-8 = 3x+9
To solve the equation 16(4^(2x-1)) = 8^(x+3), we'll need to simplify the expressions on both sides and then solve for x. Let's break down the steps:
Step 1: Simplify the expressions
On the left side, we have 16 multiplied by 4 raised to the power of (2x-1). Using the exponent properties, we can rewrite it as:
16 * (2^2)^(2x-1).
Simplifying the expression inside the parentheses, we get:
16 * 2^(4x-2).
On the right side, we have 8 raised to the power of (x+3). Using the exponent properties again, we can rewrite it as:
(2^3)^(x+3).
Simplifying the expression inside the parentheses, we get:
2^(3x+9).
Now we have the equation:
16 * 2^(4x-2) = 2^(3x+9).
Step 2: Solve for x
Since both sides of the equation have the same base (2), we can set the exponents equal to each other. This gives us:
4x-2 = 3x+9.
Step 3: Solve for x
Let's isolate the variable x by moving the 3x term to the left side and the constant term (-2) to the right side:
4x - 3x = 9 + 2.
Simplifying both sides, we have:
x = 11.
Therefore, the solution to the equation is x = 11.
To check our answer, we can substitute x = 11 back into the original equation and see if both sides are equal:
16(4^(2*11-1)) = 8^(11+3).
16(4^21) = 8^14.
Since both sides are equal, we can conclude that x = 11 is the correct solution.