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.

Division of Complex Numbers

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.

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.