In Exercises 37–52, perform the indicated operations 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 expressions 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 terms are \$20i\( and \)27i$.
5. Add the two imaginary terms together: \$20i + 27i = (20 + 27)i\(, which simplifies to \)47i\(. This is the result in standard form, where the real part is 0 and the imaginary part is \)47$.

Key Concepts

Imaginary Numbers and Complex Numbers

Imaginary numbers arise from the square roots of negative numbers, defined using the imaginary unit i, where i² = -1. Complex numbers combine real and imaginary parts in the form a + bi, allowing operations involving roots of negative numbers to be expressed in standard form.

Simplifying Square Roots of Negative Numbers

To simplify the square root of a negative number, separate it into the square root of the positive part times the square root of -1. For example, √-16 = √16 × √-1 = 4i. This process converts the expression into a form involving imaginary units, facilitating further arithmetic.

Addition of Complex Numbers and Standard Form

Adding complex numbers involves combining their real parts and imaginary parts separately. The standard form of a complex number is a + bi, where a and b are real numbers. Writing results in this form ensures clarity and consistency in representing complex expressions.