Final answer:
To find b, we need to analyze the constant term in the expanded expression and use the binomial theorem. By solving the equation (b/x)^8 = 256, we can determine that b = 2x.
Step-by-step explanation:
Step 1:
Expand the expression using the binomial theorem.
(2x^3 + b/x)^8 = 256x^24 + 3072x^20 + ... + kx^0
Step 2:
Apply the binomial theorem to find the coefficients of each term.
Step 3:
Look for the term that does not contain any variable, which is the constant term. In this case, it is
256x^24
. The exponent of x in this term is 24, so we know that 2x to the power of 3 should be raised to the power of 8 to get the exponent 24. That means 3 * 8 = 24. Therefore, b/x should be raised to the power of 8 to get the constant term 256.
Using this information, we can write the equation:
(b/x)^8 = 256
Step 4:
Solve for b. Taking the eighth root of both sides, we find:
b/x = 2
Multiplying both sides by x, we get:
b = 2x