Final answer:
To determine the value of x if (√7-xi) ÷ (1-√7i) is a real number, we can simplify the expression and solve for x, which is approximately 0.377.
Step-by-step explanation:
To determine the value of x if (√7-xi) ÷ (1-√7i) is a real number, we can simplify the expression as follows:
The expression (√7-xi) ÷ (1-√7i) becomes [(√7-xi) ÷ (1-√7i)] * [(1+√7i) ÷ (1+√7i)]. This is done to rationalize the denominator.
By multiplying the numerator and denominator using the complex conjugate of the denominator, we can simplify the expression to [(√7-xi)(1+√7i)] ÷ [1-7i].
We can further expand the expression as (7-xi*(√7i)) ÷ (1-7i), which simplifies to (7-(√7xi+i^2)) ÷ (1-7i).
Since i^2 is equal to -1, the simplified expression becomes (7-(√7xi-1)) ÷ (1-7i).
To find the real value of the expression, we equate the imaginary parts to zero.
Therefore, √7xi-1=0. By solving this equation, we find that x=1 / √7 or approximately 0.377.