In Exercises 65–70, perform the indicated operation(s) and write the result in standard form. 5√-16 + 3√-81

Verified step by step guidance

1. Step 1: Recognize that the square root of a negative number involves imaginary numbers. Recall that √(-a) = i√a, where 'i' is the imaginary unit (i.e., i² = -1).
2. Step 2: Simplify each term separately. For the first term, 5√-16, rewrite it as 5 * √16 * i, since √-16 = i√16.
3. Step 3: Simplify √16 to 4, so the first term becomes 5 * 4 * i, or 20i.
4. Step 4: For the second term, 3√-81, rewrite it as 3 * √81 * i, since √-81 = i√81.
5. Step 5: Simplify √81 to 9, so the second term becomes 3 * 9 * i, or 27i. Combine the two terms (20i + 27i) to express the result in standard form as a single imaginary number.

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. Understanding complex numbers is essential for performing operations involving square roots of negative numbers, as they allow us to express these roots in a meaningful way.

Square Roots of Negative Numbers

The square root of a negative number is not defined within the set of real numbers, but it can be expressed using imaginary numbers. For example, √-16 can be rewritten as 4i, since √-16 = √(16) * √(-1) = 4i. This concept is crucial for simplifying expressions that involve square roots of negative values.

Standard Form of Complex Numbers

The standard form of a complex number is typically written as a + bi, where 'a' and 'b' are real numbers. When performing operations with complex numbers, such as addition or multiplication, it is important to combine like terms and express the result in this standard form to clearly identify the real and imaginary components.