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

Verified step by step guidance

1. Recognize that the square root of a negative number involves the imaginary unit $i$, where $i=\sqrt{-1}$.
2. Express $\sqrt{-10}$ as $\sqrt{-1\times 10}$.
3. Use the property of square roots that $\sqrt{a\times b}=\sqrt{a}\times \sqrt{b}$.
4. Separate the expression into $\sqrt{-1}\times \sqrt{10}$.
5. Replace $\sqrt{-1}$ with $i$, resulting in $i\times \sqrt{10}$.

Key Concepts

Imaginary Numbers

Imaginary numbers are defined as multiples of the imaginary unit 'i', where i is the square root of -1. They are used to extend the real number system to solve equations that do not have real solutions, such as x^2 + 1 = 0. In this context, imaginary numbers allow us to express the square roots of negative numbers.

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part, typically expressed in the form a + bi, where a is the real part and b is the coefficient of the imaginary part. They are essential in various fields of mathematics and engineering, providing a comprehensive way to represent and manipulate numbers that include both real and imaginary components.

Square Roots of Negative Numbers

The square root of a negative number cannot be expressed as a real number, as no real number squared gives a negative result. Instead, we use imaginary numbers to represent these square roots. For example, √-10 can be rewritten as √10 * √-1, which simplifies to √10 * i, illustrating how we can express negative square roots in terms of imaginary numbers.