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Chapter 2: Problem 43

Perform the indicated operations and write the result in standard form. $$(3 \sqrt{-5})(-4 \sqrt{-12})$$

Short Answer

Expert verified
The result of the operations in standard form is \(0 - 24i\sqrt{15}\)

Step by step solution

01

Perform the multiplication

Begin by performing the multiplication, keeping the rules in mind. Here, we can multiply the coefficients, and the same for the values under the root: \((3 \sqrt{-5})(-4 \sqrt{-12}) = -12\sqrt{(-5)(-12)}\). This results in \(-12\sqrt{60}\). However, we also know that the square root of a negative number is equal to the square root of the absolute value of the number times i. Therefore we rewrite our result as \(-12i\sqrt{60}\).
02

Simplify the radical

Next, we should simplify the radical. The number 60 can be simplified to \(\sqrt{4*15}\), which further simplifies to \(2\sqrt{15}\). Thus, our expression becomes \(-12i*2\sqrt{15} = -24i\sqrt{15}\).
03

Write the result in standard form

Finally, by definition, the standard form of a complex number is written as a + bi, where a is the real part and b is the imaginary part. In our case, a equals 0 and b equals -24\sqrt{15}. So, the result in standard form is \(0 - 24i\sqrt{15}\).

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