Tính z=(〖(1+i)〗^10-1)/(〖(1+i)〗^10+1)

It looks like the given complex number is

z = ((1 + i )¹⁰ - 1) / ((1 + i )¹⁰ + 1)

Let w = 1 + i, so we rewrite

z = (w ¹⁰ - 1) / (w ¹⁰ + 1)

Since w * = 1 - i, we get w w * = (1 + i ) (1 - i ) = 1 - (-1) = 2. Multiply z by ((w *)¹⁰ + 1) / ((w *)¹⁰ + 1). This gives

z = (2¹⁰ - 1 + w ¹⁰ - (w *)¹⁰) / (2¹⁰ + 1 + w ¹⁰ + (w *)¹⁰)

Now, writing w in polar form yields

w = √2 exp(i π/4)

==>   w ¹⁰ = (√2)¹⁰ exp(i 10π/4) = 2⁵ exp(i 5π/2) = 2⁵ i

Similarly,

w * = √2 exp(-i π/4)

==>   (w *)¹⁰ = -2⁵ i

So we have

w ¹⁰ + (w *)¹⁰ = 0

w ¹⁰ - (w *)¹⁰ = 2 × 2⁵ i = 2⁶ i

Then we end up with

z = (2¹⁰ - 1 + 2⁶ i ) / (2¹⁰ + 1) = (1023 + 64i ) / 1025

89

step-by-step explanation:

answer set (b) 0.5, 3/4 80% is in order from least to greatest.  

step-by-step explanation:

in order to compare rational numbers to each other, it is much easier if they are in the same form.   typically, the best form in which to evaluate rational numbers is in decimals.   in order to convert a percent to a decimal, you would divide the given percentage by 100.   you can also accomplish the conversion by moving the decimal point over two places to the left.   for example, the decimal value of 80% is 80/100, or 0.8.   in order to convert a fraction to a decimal, you need to divide the numerator by the denominator.   for example, 3/4 is 0.75.   so, for choice (b) 0.5 3/4 (0.75), 80% (0.8) is the only choice in which the numbers are in order from least to greatest.