Question

Z=(2-i)^3:(i+2)^2+2i+3

Answer 1

Ответ этого уравнения = 56

Answer 2

To simplify the expression, let's start by expanding the numerator and the denominator.

The numerator is given by: (2 - i)^3 = (2 - i)(2 - i)(2 - i)

Expanding the numerator: (2 - i)(2 - i) = 4 - 2i - 2i + i^2 = 4 - 4i + i^2

Now, multiply (4 - 4i + i^2) by (2 - i): (4 - 4i + i^2)(2 - i) = 8 - 8i + 2i - 2i^2 - 4i + 4i^2 + i^3 = 8 - 10i - 2i^2 + i^3 = 8 - 10i - 2(-1) + (-i) = 8 - 10i + 2 + (-i) = 10 - 11i

So, the numerator simplifies to 10 - 11i.

Now, let's expand the denominator: (i + 2)^2 = (i + 2)(i + 2)

Expanding the denominator: (i + 2)(i + 2) = i^2 + 2i + 2i + 4 = -1 + 4i + 4 = 3 + 4i

So, the denominator simplifies to 3 + 4i.

Now, let's substitute the simplified numerator and denominator back into the original expression:

Z = (10 - 11i) / (3 + 4i) + 2i + 3

To simplify further, let's multiply the numerator and denominator by the conjugate of the denominator, which is (3 - 4i):

Z = ((10 - 11i) * (3 - 4i)) / ((3 + 4i) * (3 - 4i)) + 2i + 3

Expanding the numerator and the denominator:

Numerator: (10 - 11i) * (3 - 4i) = 30 - 40i - 33i + 44i^2 = 30 - 73i - 44 = -14 - 73i

Denominator: (3 + 4i) * (3 - 4i) = 9 - 12i + 12i - 16i^2 = 9 - 16i^2 = 9 - 16(-1) = 9 + 16 = 25

Substituting the simplified numerator and denominator back into the expression:

Z = (-14 - 73i) / 25 + 2i + 3

Finally, we can simplify the expression further:

Z = (-14 - 73i) / 25 + (2i * 25) / 25 + (3 * 25) / 25

Z = (-14 - 73i + 50i + 75) / 25

Z = (61 - 23i) / 25

Therefore, the simplified expression is Z = (61 - 23i) / 25.