Find each product or quotient. Simplify the answers. √-13 * √-13

Verified step by step guidance

1. Recognize that the expression involves the product of two square roots of negative numbers: \(\sqrt{-13} \times \sqrt{-13}\).
2. Recall that \(\sqrt{-a} = \sqrt{a} \times i\) where \(i\) is the imaginary unit with the property \(i^2 = -1\).
3. Rewrite each square root using the imaginary unit: \(\sqrt{-13} = \sqrt{13} \times i\).
4. Substitute back into the product: \((\sqrt{13} \times i) \times (\sqrt{13} \times i)\).
5. Use properties of multiplication to combine terms: \((\sqrt{13} \times \sqrt{13}) \times (i \times i) = 13 \times i^2\), then simplify using \(i^2 = -1\).

Key Concepts

Imaginary Numbers

Imaginary numbers extend the real number system by including the square root of negative one, denoted as i, where i² = -1. This allows for the definition and manipulation of square roots of negative numbers, such as √-13, which can be expressed as i√13.

Properties of Square Roots

The product of square roots can be simplified using the property √a * √b = √(a*b), provided a and b are non-negative. When dealing with negative numbers under the root, this property is applied carefully by factoring out the imaginary unit i.

Simplifying Products of Imaginary Numbers

When multiplying imaginary numbers, use the fact that i² = -1 to simplify expressions. For example, (√-13)(√-13) equals (i√13)(i√13) = i² * 13 = -13, converting the product back into a real number.