Write each number as the product of a real number and i. √-25

Verified step by step guidance

1. Recognize that the expression involves the square root of a negative number, which can be rewritten using the imaginary unit \(i\), where \(i = \sqrt{-1}\).
2. Rewrite \(\sqrt{-25}\) as \(\sqrt{25 \times -1}\) to separate the positive and negative parts under the square root.
3. Use the property of square roots to write \(\sqrt{25 \times -1}\) as \(\sqrt{25} \times \sqrt{-1}\).
4. Calculate \(\sqrt{25}\), which is a real number, and replace \(\sqrt{-1}\) with \(i\).
5. Express the original number as the product of the real number found in the previous step and \(i\), in the form \(a \times i\) where \(a\) is a real number.

Key Concepts

Imaginary Unit (i)

The imaginary unit 'i' is defined as the square root of -1, which is not a real number. It allows us to express the square roots of negative numbers in terms of 'i', enabling the extension of the real number system to complex numbers.

Square Roots of Negative Numbers

The square root of a negative number can be rewritten using the imaginary unit 'i'. For example, √-25 can be expressed as √25 × √-1, which simplifies to 5i, showing how negative radicands are handled in algebra.

Product of a Real Number and i

Any number involving the square root of a negative number can be expressed as the product of a real number and 'i'. This form separates the real magnitude from the imaginary unit, making it easier to work with complex numbers in algebraic operations.