Evaluate each expression. (-4)^1/2

Verified step by step guidance

1. Recognize that the expression \((-4)^{1/2}\) represents the square root of -4, since an exponent of \(\frac{1}{2}\) means taking the square root.
2. Recall that the square root of a negative number is not a real number, but an imaginary number involving \(i\), where \(i = \sqrt{-1}\).
3. Rewrite \((-4)^{1/2}\) as \(\sqrt{-4} = \sqrt{4 \times (-1)}\).
4. Use the property of square roots to separate the factors: \(\sqrt{4 \times (-1)} = \sqrt{4} \times \sqrt{-1}\).
5. Evaluate \(\sqrt{4} = 2\) and \(\sqrt{-1} = i\), so the expression simplifies to \$2i$.

Key Concepts

Rational Exponents

Rational exponents represent roots and powers simultaneously. For example, an exponent of 1/2 means the square root of the base. Understanding how to interpret and manipulate rational exponents is essential for evaluating expressions like (-4)^(1/2).

Square Roots of Negative Numbers

The square root of a negative number is not a real number but an imaginary number. Specifically, the square root of -1 is defined as i, the imaginary unit. This concept is crucial when evaluating expressions like (-4)^(1/2), which involves the square root of a negative number.

Complex Numbers

Complex numbers extend the real number system to include imaginary numbers, expressed as a + bi, where i is the imaginary unit. Understanding complex numbers allows one to work with and simplify expressions involving roots of negative numbers, such as (-4)^(1/2) = 2i.