In Exercises 29–36, simplify and write the result in standard form.

√−196

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Recognize that the expression involves the square root of a negative number, which means we are dealing with imaginary numbers in the complex number system.

Recall the definition of the imaginary unit: \(i = \sqrt{-1}\), so \(\sqrt{-a} = i\sqrt{a}\) for any positive real number \(a\).

Rewrite the expression \(\sqrt{-196}\) as \(\sqrt{-1 \times 196}\), which can be separated into \(\sqrt{-1} \times \sqrt{196}\).

Substitute \(\sqrt{-1}\) with \(i\) and simplify \(\sqrt{196}\) by finding its positive square root.

Express the final answer in standard form for complex numbers, which is \(a + bi\), where \(a\) and \(b\) are real numbers.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Imaginary Numbers

Imaginary numbers extend the real number system by including the square root of negative one, denoted as i. This allows for the definition of the square root of negative numbers, such as √−196, which can be expressed as a multiple of i.

Simplifying Square Roots

Simplifying square roots involves factoring the radicand into perfect squares and other factors. For example, √196 is simplified to 14 because 196 is 14 squared. This process helps in rewriting expressions in their simplest form.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers and i is the imaginary unit. Writing results in this form clearly separates the real and imaginary parts, which is essential for further operations and interpretations.

In Exercises 29–36, simplify and write the result in standard form.√−196