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 roots of negative numbers involve imaginary numbers. Recall that \(\sqrt{-a} = \sqrt{a} \times i\), where \(i\) is the imaginary unit with the property \(i^2 = -1\).

Rewrite each term using the imaginary unit: \(\sqrt{-64} = \sqrt{64} \times i\) and \(\sqrt{-25} = \sqrt{25} \times i\).

Calculate the square roots of the positive numbers: \(\sqrt{64} = 8\) and \(\sqrt{25} = 5\).

Substitute these values back into the expression: \(8i - 5i\).

Combine like terms (both are imaginary terms) to write the result in standard form \(a + bi\), where \(a\) and \(b\) are real numbers.

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주요 개념

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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.

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