Problem 43 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The product of the complex numbers is \(24\sqrt{15}\)

Step 1: Identify Components of Complex Numbers

Identify the real and imaginary components of the two complex numbers. The first complex number has a real component of 3 and an imaginary component of \(\sqrt{-5}\). The second complex number has a real component of -4 and an imaginary component of \(\sqrt{-12}\).

Step 2: Simplifying the Imaginary Components

Simplify the imaginary components of the complex numbers by taking out the negative under the square root which will turn into 'i'. This gives the first complex number as \(3i\sqrt{5}\) and the second number as \(-4i\sqrt{12}\).

Step 3: Perform Multiplication

Multiply the real components together and multiply imaginary components together to get \((-12i^2)(\sqrt{60})\). The \(i^2\) factor is equal to -1 (since \(i^2 = -1\)), and \(\sqrt{60}\) can be simplified to \(2\sqrt{15}\). So this simplifies to \(12*2\sqrt{15}\) which equals to \(24\sqrt{15}\).

Step 4: Writing in Standard Form

Since the solution is a real number, standard form is the final simplified real number. The multiplication of the two complex numbers yields a real number.