Write each number as the product of a real number and i. √-10

Verified step by step guidance

1. Recall that the imaginary unit \(i\) is defined as \(i = \sqrt{-1}\).
2. Express the square root of a negative number in terms of \(i\): \(\sqrt{-10} = \sqrt{10 \times (-1)}\).
3. Use the property of square roots to separate the factors: \(\sqrt{10 \times (-1)} = \sqrt{10} \times \sqrt{-1}\).
4. Substitute \(\sqrt{-1}\) with \(i\): \(\sqrt{10} \times i\).
5. Therefore, \(\sqrt{-10}\) can be written as the product of the real number \(\sqrt{10}\) and \(i\).

Key Concepts

Imaginary Unit (i)

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

Simplifying Square Roots of Negative Numbers

To simplify the square root of a negative number, separate it into the square root of the negative part and the positive part, such as √(-a) = √(-1) × √a = i√a, where a is positive.

Expressing Complex Numbers in Standard Form

Complex numbers are often written as a product of a real number and i when dealing with purely imaginary numbers. For example, √-10 can be expressed as (√10) × i, showing the real magnitude and the imaginary unit.