Question

Решить пример 2-i/5 : 2+i/10

Answer 1

$2 - \frac{i}{5 } \times \frac{1}{2} + \frac{i}{10} \\ \ 2 - \frac{i}{10} + \frac{i}{10 } \\ 2$
Ответ:2

Answer 2

To solve the expression, you need to rationalize the denominators first. Rationalizing the denominators involves multiplying both the numerator and denominator of each fraction by the conjugate of the denominator. The conjugate of a complex number of the form a + bi is a - bi.

Let's solve the expression step by step:

Step 1: Rationalize the denominators For the first fraction: (2 - i) / 5 For the second fraction: (2 + i) / 10

Step 2: Combine the fractions (2 - i) / 5 : (2 + i) / 10

Step 3: Invert the second fraction and multiply (2 - i) / 5 * 10 / (2 + i)

Step 4: Multiply the numerators and denominators (2 - i) * 10 / (5 * (2 + i))

Step 5: Distribute and simplify the numerator (20 - 10i) / (10 + 5i)

Step 6: Factor out 5 from the numerator and denominator (5 * (4 - 2i)) / (5 * (2 + i))

Step 7: Cancel out the common factor (4 - 2i) / (2 + i)

Now, if needed, you can simplify it further. One common practice is to multiply the fraction by the conjugate of the denominator to get rid of the imaginary part in the denominator:

Step 8: Rationalize the denominator again (4 - 2i) / (2 + i) * (2 - i) / (2 - i)

Step 9: Multiply the numerators and denominators (8 - 6i + 2i - 2i^2) / (4 - 2i + 2i - i^2)

Step 10: Simplify using i^2 = -1 (8 - 4i - 2) / (4 + 1)

Step 11: Simplify further (6 - 4i) / 5

So, the final simplified expression is: (6 - 4i) / 5