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 problem involves square roots of negative numbers. These are imaginary numbers, and we use the imaginary unit i, where i = √-1.
2. Rewrite each term using the property of square roots for negative numbers: √-a = i√a. For the first term, rewrite 5√-16 as 5i√16. For the second term, rewrite 3√-81 as 3i√81.
3. Simplify the square roots of the positive numbers. For √16, the result is 4, and for √81, the result is 9. This gives 5i√16 = 5i(4) = 20i and 3i√81 = 3i(9) = 27i.
4. Combine the imaginary terms. Add 20i and 27i together to get (20 + 27)i.
5. Write the final result in standard form for complex numbers, which is a + bi. Since there is no real part in this problem, the result is purely imaginary and can be written as 47i.

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 is 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 and presented in this format. This helps in clearly identifying the real and imaginary components of the number.

Operations with Complex Numbers

Operations with complex numbers include addition, subtraction, multiplication, and division. When performing these operations, it is essential to combine like terms, particularly separating the real and imaginary parts. For instance, when adding two complex numbers, you add their real parts together and their imaginary parts together to form a new complex number.