Write each number as the product of a real number and i. -√-18

Verified step by step guidance

1. Recognize that the expression involves the square root of a negative number, which can be rewritten using the imaginary unit \(i\), where \(i = \sqrt{-1}\).
2. Rewrite the expression \(\sqrt{-18}\) as \(\sqrt{18} \times \sqrt{-1}\), which becomes \(\sqrt{18} \times i\).
3. Simplify \(\sqrt{18}\) by factoring it into \(\sqrt{9 \times 2}\), which equals \(\sqrt{9} \times \sqrt{2}\).
4. Calculate \(\sqrt{9}\) as 3, so \(\sqrt{18}\) simplifies to \(3\sqrt{2}\).
5. Combine the results to express the original number as the product of a real number and \(i\): \(3\sqrt{2} \times i\).

Key Concepts

Imaginary Unit (i)

The imaginary unit i is defined as the square root of -1, i.e., i² = -1. It allows us to express the square roots of negative numbers in terms of real numbers multiplied by i, enabling the extension of the real number system to complex numbers.

Simplifying Square Roots of Negative Numbers

To simplify the square root of a negative number, separate it into the square root of the negative sign and the square root of the positive part. For example, √(-18) = √(-1) × √(18) = i × √(18), which can then be simplified further by factoring.

Prime Factorization and Simplification of Radicals

Simplifying radicals involves breaking down the number inside the square root into prime factors and extracting perfect squares. For √18, since 18 = 9 × 2, √18 = √9 × √2 = 3√2, which helps express the number in simplest radical form.