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 square roots of negative numbers involve 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 using the imaginary unit: \(\sqrt{-64} = \sqrt{64} \times i\) and \(\sqrt{-25} = \sqrt{25} \times i\).
3. Calculate the square roots of the positive numbers: \(\sqrt{64} = 8\) and \(\sqrt{25} = 5\).
4. Substitute these values back into the expression: \$8i - 5i$.
5. Combine like terms (both are imaginary terms) to write the result in standard form \(a + bi\), where \(a\) and \(b\) are real numbers.

Key Concepts

Imaginary Numbers and the Imaginary Unit i

Imaginary numbers arise when taking the square root of negative numbers. The imaginary unit i is defined as √−1, allowing us to express roots of negative numbers as multiples of i, such as √−64 = 8i.

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 and the imaginary unit i. For example, √−25 = √25 × √−1 = 5i, which helps in performing arithmetic operations.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers. After performing operations with imaginary numbers, express the result in this form to clearly separate the real and imaginary parts.