Simplify and write the result in standard form. √-49

Verified step by step guidance

1. Recognize that the square root of a negative number involves imaginary numbers. The imaginary unit, denoted as 'i', is defined as the square root of -1, i.e., i = √-1.
2. Rewrite the given expression √-49 as √(-1 × 49).
3. Use the property of square roots that √(a × b) = √a × √b to separate the terms: √(-1 × 49) = √-1 × √49.
4. Substitute √-1 with 'i' and simplify √49 to 7 (since 49 is a perfect square): √-1 × √49 = i × 7.
5. Write the final simplified expression in standard form for complex numbers: 7i.

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. Understanding complex numbers is essential for simplifying expressions involving square roots of negative numbers.

Imaginary Unit

The imaginary unit 'i' is defined as the square root of -1. It allows for the extension of the real number system to include solutions to equations that do not have real solutions, such as x^2 + 1 = 0. In the context of the question, √-49 can be rewritten using 'i' to express the result in terms of complex numbers.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where 'a' and 'b' are real numbers. When simplifying expressions involving square roots of negative numbers, it is important to express the result in this form to clearly indicate both the real and imaginary components. For example, √-49 simplifies to 7i, which is in standard form.