In Exercises 37–52, perform the indicated operations and write the result in standard form.
( −6 − √(−12)) / 48

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Identify the expression to simplify: \(\frac{-5 - \sqrt{-12}}{48}\). Notice that \(\sqrt{-12}\) involves an imaginary number because of the negative under the square root.

Rewrite the square root of the negative number using the imaginary unit \(i\), where \(i = \sqrt{-1}\). So, \(\sqrt{-12} = \sqrt{12} \times i\).

Simplify \(\sqrt{12}\) by factoring it into \(\sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\). Therefore, \(\sqrt{-12} = 2\sqrt{3}i\).

Substitute back into the original expression: \(\frac{-5 - 2\sqrt{3}i}{48}\). Now, separate the real and imaginary parts by dividing both terms in the numerator by 48.

Write the expression in standard form \(a + bi\) as \(\frac{-5}{48} - \frac{2\sqrt{3}}{48}i\). Simplify the fractions if possible to complete the standard form.

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주요 개념

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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 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, where √(-x) = i√x. Recognizing and converting these roots into terms involving i is crucial for handling expressions like √(-12), enabling further algebraic manipulation.

Operations with Complex Numbers

Performing addition, subtraction, multiplication, or division with complex numbers requires combining like terms and applying algebraic rules, including distributing and rationalizing denominators. Mastery of these operations ensures the expression can be simplified correctly into standard form.

In Exercises 37–52, perform the indicated operations and write the result in standard form. ( −6 − √(−12)) / 48