In Exercises 29–36, simplify and write the result in standard form. ____ √−108

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Recognize that the expression involves the square root of a negative number, which means the result will be a complex number. Recall that \(\sqrt{-a} = \sqrt{a} \times i\), where \(i\) is the imaginary unit with the property \(i^2 = -1\).

Rewrite the expression \(\sqrt{-108}\) as \(\sqrt{108} \times i\) to separate the imaginary unit from the real number under the root.

Simplify \(\sqrt{108}\) by factoring 108 into its prime factors or perfect squares. For example, \(108 = 36 \times 3\), and since \(\sqrt{36} = 6\), you can write \(\sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3}\).

Substitute back to get the expression in terms of \(i\): \(\sqrt{-108} = 6\sqrt{3} \times i\).

Write the final answer in standard form for complex numbers, which is \(a + bi\). Since there is no real part here, the expression is \(0 + 6\sqrt{3}i\).

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주요 개념

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Simplifying Square Roots of Negative Numbers

When simplifying the square root of a negative number, recognize that it involves imaginary numbers. The square root of a negative number can be expressed as the product of the imaginary unit 'i' (where i² = -1) and the square root of the corresponding positive number.

Prime Factorization for Simplifying Radicals

To simplify a square root, break down the number inside the root into its prime factors. Pair factors to extract them from under the root, simplifying the expression. For example, √108 can be factored into √(36 × 3) = 6√3.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where 'a' is the real part and 'b' is the imaginary coefficient. After simplifying the radical, express the result in this form to clearly separate real and imaginary components.