Answer:
-√6(cos(π+π/6)+i sin(π+π/6))
or
-2.449(cos(3.658)+i sim( 3.658)
Explanation:
First, let's simplify the expression in the denominator:
4√2(cos(5π/6)+i sin(5π/6)) = 4√2(√3/2 + i/2) = 2√6 + 2i√2
Now we can rewrite the entire expression as:
32(cos(7π/4)+i sin(7π/4)) / (2√6 + 2i√2)
To simplify the division, we can multiply both the numerator and denominator by the conjugate of the denominator, which is:
2√6 - 2i√2
Multiplying the numerator and denominator by this conjugate, we get:
32(cos(7π/4)+i sin(7π/4)) * (2√6 - 2i√2) / [(2√6 + 2i√2) * (2√6 - 2i√2)]
Simplifying the denominator on the right side gives us:
(2√6)^2 - (2i√2)^2 = 24 - 8 = 16
So we can simplify the expression further to:
32(cos(7π/4)+i sin(7π/4)) * (2√6 - 2i√2) / 16
Canceling out the 16 in the denominator and simplifying the numerator gives:
2(cos(7π/4)+i sin(7π/4)) * (√6 - i√2)
Now we can expand the complex multiplication in the numerator:
2(cos(7π/4)√6 - sin(7π/4)√2 + i(cos(7π/4)√2 + sin(7π/4)√6))
Simplifying the trigonometric functions using the values of sine and cosine for 7π/4 (which is equivalent to -π/4), we get:
2(-√6/2 - √2/2i + i√2/2 - √6/2i)
Simplifying further, we can combine like terms and rationalize the denominator:
-√6 - √3i
Therefore, the final answer is:
-√6(cos(π+π/6)+i sin(π+π/6))
Hope this helps!