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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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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