Problem 31 - Algebra and Trigonometry

Question

Perform the indicated operations and write the result in standard form. $$ 5 \sqrt{-16}+3 \sqrt{-81} $$

Step-by-Step Solution

Answer

The result of the given operation in standard form is 47i.

Step 1: Understand the concept of square root of a negative number

Recall that the square root of -1 is defined as the imaginary unit 'i'. That is, \( \sqrt{-1} = i \) . So, in general, \( \sqrt{-x} = i\sqrt{x} \) where x is a positive real number.

Step 2: Calculate the square roots

Now, calculate the square roots in the expression: 5 \(\sqrt{-16}\) + 3 \(\sqrt{-81}\). By applying the concept from Step 1, we get: 5 * i \(\sqrt{16}\) + 3 * i \(\sqrt{81}\). After calculating the square roots, we get: 5 * 4i + 3 * 9i.

Step 3: Simplify the expressions

Simplify each of the terms in the expression: 20i + 27i.

Step 4: Add the terms

Finally, add the two terms together: 20i + 27i = 47i.