Perform the indicated operations and write the result in standard form. (4 + √−8 )/ 2

검증된 단계별 안내

Identify the expression to simplify: \(\frac{4 + \sqrt{-8}}{2}\).

Recall that \(\sqrt{-8}\) can be rewritten using imaginary unit \(i\) as \(\sqrt{-8} = \sqrt{8} \cdot i\).

Simplify \(\sqrt{8}\) by expressing it as \(\sqrt{4 \times 2} = 2\sqrt{2}\), so \(\sqrt{-8} = 2\sqrt{2}i\).

Substitute back into the original expression: \(\frac{4 + 2\sqrt{2}i}{2}\).

Separate the fraction into real and imaginary parts: \(\frac{4}{2} + \frac{2\sqrt{2}i}{2}\), then simplify each term to write the expression in standard form \(a + bi\).

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Complex Numbers and Standard Form

Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. The standard form requires separating the real and imaginary components clearly, which is essential when simplifying expressions involving square roots of negative numbers.

Simplifying Square Roots of Negative Numbers

The square root of a negative number involves imaginary units, defined as i = √−1. For example, √−8 can be rewritten as √8 × i, which simplifies to 2√2 i. Recognizing this allows conversion of complex radicals into standard complex number form.

Operations with Complex Numbers

Performing addition, subtraction, multiplication, or division with complex numbers requires combining like terms (real with real, imaginary with imaginary). When dividing, rationalizing the denominator by multiplying numerator and denominator by the conjugate is often necessary to express the result in standard form.