In Exercises 21–28, divide and express the result in standard form. 5i/(2 - i)

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Rewrite the division problem as a fraction: $ \frac{5i}{2 - i} $. The goal is to simplify this expression and express it in standard form $ a + bi $, where $ a $ and $ b $ are real numbers.

To simplify, multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of $ 2 - i $ is $ 2 + i $. Multiply: $ \frac{5i}{2 - i} \cdot \frac{2 + i}{2 + i} $.

Expand the numerator $ 5i(2 + i) $ using the distributive property: $ 5i \cdot 2 + 5i \cdot i = 10i + 5i^2 $. Recall that $ i^2 = -1 $, so substitute $ 5i^2 $ with $ -5 $. The numerator becomes $ 10i - 5 $.

Expand the denominator $ (2 - i)(2 + i) $ using the difference of squares formula: $ a^2 - b^2 $. Here, $ a = 2 $ and $ b = i $, so $ (2 - i)(2 + i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5 $.

Combine the simplified numerator and denominator: $ \frac{10i - 5}{5} $. Separate the terms in the fraction: $ \frac{10i}{5} - \frac{5}{5} $. Simplify each term to express the result in standard form $ a + bi $.

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Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part and b is the coefficient of the imaginary unit i, which is defined as the square root of -1. Understanding complex numbers is essential for performing operations such as addition, subtraction, multiplication, and division.

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Dividing complex numbers involves multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi. This process eliminates the imaginary part in the denominator, allowing the result to be expressed in standard form, which is a + bi.

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Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers. In this form, a represents the real part and b represents the imaginary part. Expressing complex numbers in standard form is crucial for clarity and consistency in mathematical communication, especially when performing further calculations or comparisons.

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