Final answer:
To find the value of ∛x + x² when log_1/2 x = 3, solve for x to get 1/8. Then calculate the cube root of x and x², and add them together to get the final answer, which is 33/64.
Step-by-step explanation:
If log_1/2 x = 3, we want to find the value of ∛x + x². First, let's solve for x using the definition of logarithms: x = (1/2)³.
Since 1/2 to the power of 3 is 1/8, x equals 1/8. Now, let's compute the cube root of x, which is ∛(1/8) or 1/2, and x², which is (1/8)² or 1/64.
To find the sum ∛x + x², we add the cube root of x and the square of x: 1/2 + 1/64. To combine these fractions, we find a common denominator, which is 64, so we can express ∛x as 32/64 and x² as 1/64. The sum is therefore (32/64) + (1/64) = 33/64.
The final answer is 33/64.