Find each square root. See Example 1. √-121

Verified step by step guidance

1. Recognize that the square root of a negative number involves imaginary numbers because the square root of a negative number is not defined in the set of real numbers.
2. Recall the definition of the imaginary unit \(i\), where \(i = \sqrt{-1}\), so \(\sqrt{-a} = i\sqrt{a}\) for any positive real number \(a\).
3. Rewrite the expression \(\sqrt{-121}\) as \(\sqrt{-1 \times 121}\) to separate the negative sign from the positive number.
4. Apply the property of square roots to write \(\sqrt{-121} = \sqrt{-1} \times \sqrt{121}\).
5. Substitute \(\sqrt{-1}\) with \(i\) and simplify \(\sqrt{121}\) to get the expression in terms of \(i\) and a real number.

Key Concepts

Imaginary Numbers

Imaginary numbers extend the real number system to include the square roots of negative numbers. The imaginary unit 'i' is defined such that i² = -1, allowing us to express √-121 as 11i.

Square Roots of Negative Numbers

The square root of a negative number cannot be found within the real numbers. Instead, it is expressed using imaginary numbers, where √-a = √a × i for any positive real number a.

Simplifying Square Roots

Simplifying square roots involves factoring the number under the root into perfect squares and other factors. For example, √121 = 11, so √-121 = 11i by combining the imaginary unit with the simplified root.