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

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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).

Rewrite each square root using the imaginary unit: √(-64) = i√64 and √(-25) = i√25.

Simplify the square roots of the positive numbers: √64 = 8 and √25 = 5. Therefore, √(-64) = 8i and √(-25) = 5i.

Perform the subtraction of the imaginary terms: 8i - 5i.

Combine the imaginary terms to write the result in standard form (a + bi), where 'a' is the real part and 'b' is the imaginary part.

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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 extend the number system beyond real numbers.

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) = 8 * i. This concept is crucial for solving problems that involve square roots of negative values, as it leads to the use of complex numbers.

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 subtraction, it is important to combine like terms to express the result in this standard form. This ensures clarity and consistency in representing complex numbers in mathematical expressions.