In Exercises 37–52, perform the indicated operations and write the result in standard form. ___ ___√−64 − √−25

Verified step by step guidance

1. Recognize that the problem involves square roots of negative numbers, which means we are dealing with imaginary numbers.
2. Recall that the square root of a negative number can be expressed in terms of the imaginary unit \(i\), where \(i = \sqrt{-1}\).
3. Rewrite \(\sqrt{-64}\) as \(\sqrt{64} \cdot \sqrt{-1} = 8i\).
4. Rewrite \(\sqrt{-25}\) as \(\sqrt{25} \cdot \sqrt{-1} = 5i\).
5. Subtract the two imaginary numbers: \(8i - 5i\).

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 'bi' is the imaginary part. The imaginary unit 'i' is defined as the square root of -1. Understanding complex numbers is essential for performing operations involving square roots of negative numbers, as seen in the given question.

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, √-64 can be simplified to 8i, since √-64 = √(64) * √(-1) = 8i. This concept is crucial for solving the problem, as it allows us to rewrite the square roots in terms of complex numbers.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where 'a' and 'b' are real numbers. When performing operations with complex numbers, such as addition or subtraction, it is important to combine like terms to express the result in this standard form. In the context of the question, after simplifying the square roots, the final result should be expressed in this format.