In Exercises 1–10, perform the indicated operations and write the result in standard form. ___ ___√−32 − √−18

Verified step by step guidance

1. Step 1: Recognize that both terms involve square roots of negative numbers, which means we are dealing with imaginary numbers. Recall that \( \sqrt{-1} = i \), where \( i \) is the imaginary unit.
2. Step 2: Rewrite each square root in terms of \( i \). For \( \sqrt{-32} \), express it as \( \sqrt{32} \times i \). Similarly, express \( \sqrt{-18} \) as \( \sqrt{18} \times i \).
3. Step 3: Simplify \( \sqrt{32} \) and \( \sqrt{18} \) by factoring them into their prime factors. \( 32 = 2^5 \) and \( 18 = 2 \times 3^2 \).
4. Step 4: Simplify the square roots: \( \sqrt{32} = \sqrt{2^5} = 2^2 \times \sqrt{2} = 4\sqrt{2} \) and \( \sqrt{18} = \sqrt{2 \times 3^2} = 3\sqrt{2} \).
5. Step 5: Substitute back into the expression: \( 4\sqrt{2}i - 3\sqrt{2}i \). Combine like terms to simplify the expression.

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 '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 express these roots in a meaningful way.

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, √−32 can be rewritten as √32 * √−1, which simplifies to 4√2 * i. This concept is crucial for solving problems that involve square roots of negative values, as it transitions the problem into the realm 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 (real with real and imaginary with imaginary) to express the result in this standard form. This ensures clarity and consistency in representing complex numbers.