Final answer:
To express a complex number in the form a + bi, separate the real and imaginary parts. For √i, write it as √(1 * i), giving (1/√2) + (1/√2)i. For √(21+20i), find the magnitude and argument of the complex number to express it as 29(cos(arctan(20/21)) + isin(arctan(20/21))). For √(-15+8i), find the magnitude and argument to express it as 17(cos(arctan(8/-15)) + isin(arctan(8/-15))).
Step-by-step explanation:
To express a complex number in the form a + bi, we need to separate the real and imaginary parts.
a) For √i, we can write it as √(1 * i). Taking the square root of i gives us (1/√2) + (1/√2)i.
b) For √(21 + 20i), we need to first find the magnitude and argument of the complex numb
er. The magnitude can be found using the Pythagorean theorem: √(21^2 + 20^2) = √841 = 29. The argument can be found using the inverse tangent function: arctan(20/21). So, we can express it as 29(cos(arctan(20/21)) + isin(arctan(20/21))).
c) For √(-15 + 8i), again we find the magnitude and argument of the complex number: √((-15)^2 + 8^2) = √289 = 17. The argument is arctan(8/-15). So, we can express it as 17(cos(arctan(8/-15)) + isin(arctan(8/-15))).