Write each number in standard form a+bi. -3+ √-18 / 24

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Identify the expression given: \(-3 + \frac{\sqrt{-18}}{24}\). Our goal is to write it in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit.

Simplify the square root of the negative number: \(\sqrt{-18} = \sqrt{18} \cdot \sqrt{-1} = \sqrt{18}i\). Since \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\), we have \(\sqrt{-18} = 3\sqrt{2}i\).

Substitute back into the expression: \(-3 + \frac{3\sqrt{2}i}{24}\). This can be rewritten as \(-3 + \left(\frac{3\sqrt{2}}{24}\right) i\).

Simplify the fraction \(\frac{3\sqrt{2}}{24}\) by dividing numerator and denominator by 3: \(\frac{3\sqrt{2}}{24} = \frac{\sqrt{2}}{8}\). So the expression becomes \(-3 + \frac{\sqrt{2}}{8} i\).

Now the expression is in standard form \(a + bi\) with \(a = -3\) and \(b = \frac{\sqrt{2}}{8}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 imaginary unit i satisfies i² = -1. Writing a number in standard form means separating and simplifying its real and imaginary components clearly.

Simplifying Square Roots of Negative Numbers

To simplify the square root of a negative number, factor out -1 and rewrite it using i, the imaginary unit. For example, √-18 = √(18) * √(-1) = √18 * i. This step is essential to convert expressions involving negative square roots into complex numbers.

Fraction Simplification in Complex Numbers

When a complex expression is divided by a number, simplify the numerator and denominator separately, then divide each part by the denominator. This helps to write the complex number in the form a + bi by ensuring both real and imaginary parts are expressed as simplified fractions.