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 expressions involve square roots of negative numbers, which means we are dealing with imaginary numbers. Recall that \(\sqrt{-a} = \sqrt{a} \times i\), where \(i\) is the imaginary unit with the property \(i^2 = -1\).
2. Rewrite each term by separating the negative sign inside the square root: \(5\sqrt{-16} = 5 \times \sqrt{16} \times i\) and \(3\sqrt{-81} = 3 \times \sqrt{81} \times i\).
3. Calculate the square roots of the positive numbers: \(\sqrt{16} = 4\) and \(\sqrt{81} = 9\), so the terms become \(5 \times 4 \times i\) and \(3 \times 9 \times i\) respectively.
4. Multiply the coefficients: \(5 \times 4 = 20\) and \(3 \times 9 = 27\), so the expression is now \$20i + 27i$.
5. Combine like terms (both are imaginary terms) by adding the coefficients: \$20i + 27i = (20 + 27)i\(, which simplifies to \)47i\(. This is the expression in standard form, where the real part is 0 and the imaginary part is \)47i$.

Key Concepts

Complex Numbers and Imaginary Unit

Complex numbers include a real part and an imaginary part, where the imaginary unit 'i' is defined as √−1. When dealing with square roots of negative numbers, express them in terms of 'i' to simplify and perform operations.

Simplifying Square Roots of Negative Numbers

To simplify √−a, rewrite it as √a × √−1, which equals √a × i. For example, √−16 becomes 4i. This allows you to convert roots of negative numbers into imaginary numbers for further calculation.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary part. After performing operations, express the result in this form for clarity and consistency.