Final answer:
To solve the equation 2^x * 4^(x+1) = 8^2 - 3 to find the value of x, simplify the equation by rewriting each power of 2, 4, and 8 in terms of a common base. Equate the simplified exponents and solve for x.
Step-by-step explanation:
To solve the equation 2^x * 4^(x+1) = 8^2 - 3 for x, we can simplify the equation by expressing each power of 2, 4, and 8 in terms of a common base. Since 4 is equal to 2^2, and 8 is equal to 2^3, we can rewrite the equation as:
2^x * (2^2)^(x+1) = (2^3)^2 - 3
Using the properties of exponents, we can simplify the equation as follows:
2^x * 2^(2(x+1)) = 2^(3*2) - 3
Now that the bases are the same, we can equate the exponents:
x + 2(x+1) = 3*2 - 3
Simplifying the equation further:
x + 2x + 2 = 6 - 3
Combining like terms:
3x + 2 = 3
Subtracting 2 from both sides:
3x = 1
Dividing both sides by 3:
x = 1/3