In Exercises 37–52, perform the indicated operations and write the result in standard form. ___ −8 + √−32 / 24

Verified step by step guidance

1. Identify the expression to simplify: \(-8 + \frac{\sqrt{-32}}{24}\). Notice that the square root involves a negative number, which means we will use imaginary numbers.
2. Rewrite the square root of the negative number using the imaginary unit \(i\), where \(i = \sqrt{-1}\). So, \(\sqrt{-32} = \sqrt{32} \times \sqrt{-1} = \sqrt{32}i\).
3. Simplify \(\sqrt{32}\) by expressing it as \(\sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}\). Therefore, \(\sqrt{-32} = 4\sqrt{2}i\).
4. Substitute back into the expression: \(-8 + \frac{4\sqrt{2}i}{24}\). Simplify the fraction \(\frac{4\sqrt{2}i}{24}\) by dividing numerator and denominator by 4, resulting in \(\frac{\sqrt{2}i}{6}\).
5. Write the expression in standard form \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a = -8\) and \(b = \frac{\sqrt{2}}{6}\). So the expression is \(-8 + \frac{\sqrt{2}}{6}i\).

Key Concepts

Complex Numbers and Imaginary Unit

Complex numbers consist of a real part and an imaginary part, expressed as a + bi, where i is the imaginary unit with the property i² = -1. Understanding how to represent and manipulate these numbers is essential when dealing with square roots of negative numbers.

Simplifying Square Roots of Negative Numbers

To simplify the square root of a negative number, factor out the imaginary unit i by rewriting √(-a) as i√a, where a is positive. This allows the expression to be converted into a form involving real numbers and i, facilitating further operations.

Operations with Complex Numbers in Standard Form

Standard form for complex numbers is a + bi, where a and b are real numbers. Performing addition, subtraction, multiplication, or division requires combining like terms and applying algebraic rules, including i² = -1, to simplify the expression correctly.