Decide whether each statement is true or false. If false, correct the right side of the equation. √-25 = 5i

Verified step by step guidance

1. Recall that the square root of a negative number can be expressed using the imaginary unit \(i\), where \(i = \sqrt{-1}\).
2. Rewrite the expression \(\sqrt{-25}\) as \(\sqrt{25 \times -1}\), which can be separated into \(\sqrt{25} \times \sqrt{-1}\).
3. Calculate \(\sqrt{25}\), which is 5, and recognize that \(\sqrt{-1} = i\).
4. Combine these results to express \(\sqrt{-25}\) as \$5i$.
5. Conclude that the statement \(\sqrt{-25} = 5i\) is true.

Key Concepts

Imaginary Numbers and the Imaginary Unit

Imaginary numbers extend the real number system by including the imaginary unit 'i', defined as the square root of -1. This allows for the square roots of negative numbers to be expressed in terms of 'i', such as √-25 = 5i, where 5 is the positive real number and 'i' represents the imaginary part.

Square Roots of Negative Numbers

The square root of a negative number is not a real number but an imaginary number. To find it, factor out the negative sign as √-a = √(-1) * √a = i√a, where 'a' is a positive real number. For example, √-25 = i√25 = 5i.

Evaluating and Correcting Equations Involving Imaginary Numbers

When evaluating equations with imaginary numbers, it is important to verify the equality by correctly applying the properties of 'i'. If an equation is false, identify the mistake and rewrite the right side correctly, ensuring the imaginary unit is properly included and the real part is accurate.