In Exercises 53–58, perform the indicated operation(s) and write the result in standard form. ___ ___ 5√−16 + 3√−81

Verified step by step guidance

1. Recognize that the problem involves complex numbers, as the square roots of negative numbers are not real.
2. Rewrite the square roots of negative numbers using the imaginary unit \(i\), where \(i = \sqrt{-1}\).
3. Express \(\sqrt{-16}\) as \(\sqrt{16} \cdot \sqrt{-1} = 4i\) and \(\sqrt{-81}\) as \(\sqrt{81} \cdot \sqrt{-1} = 9i\).
4. Substitute these expressions back into the original problem: \(5(4i) + 3(9i)\).
5. Simplify the expression by performing the multiplication and addition: \(20i + 27i\).

Key Concepts

Imaginary Numbers

Imaginary numbers are defined as multiples of the imaginary unit 'i', where i is the square root of -1. They arise when taking the square root of negative numbers, which is not possible within the realm of real numbers. For example, √-16 can be expressed as 4i, since √16 = 4 and the negative sign introduces the imaginary unit.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where 'a' is the real part and 'b' is the imaginary part. In the context of the given problem, after performing the indicated operations, the result should be simplified to this form. This allows for easier interpretation and manipulation of complex numbers in mathematical operations.

Addition of Complex Numbers

To add complex numbers, you combine their real parts and their imaginary parts separately. For instance, when adding 5√-16 and 3√-81, you first convert each term into its standard form and then sum the real components and the imaginary components. This process ensures that the final result is also in standard form, facilitating further calculations or interpretations.