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Chapter 2: Problem 32

Perform the indicated operations and write the result in standard form. $$5 \sqrt{-8}+3 \sqrt{-18}$$

Short Answer

Expert verified
The result in standard form is \(19i\sqrt{2}\)

Step by step solution

01

Simplify each square root separately

We rewrite the negative numbers under the square roots as the product of -1 and a positive number. Thus the expression becomes \(5 \sqrt{-1*8} + 3 \sqrt{-1*18}\). We then simplify the square roots noting the fact that \(\sqrt{-1}\) is equivalent to 'i' in complex numbers. Hence the expression becomes \(5*2i\sqrt{2} + 3*3i\sqrt{2}\).
02

Simplify the expression further

In this step, we multiply the numbers in front of 'i' for each term. This results to \(10i\sqrt{2} + 9i\sqrt{2}\).
03

Combine like terms

Both simplified square roots are equal, hence they are like terms. The resulting expression after combining them is \((10i + 9i)\sqrt{2}\). Thus, obtaining \(19i\sqrt{2}\). We usually write complex numbers of the form 'a + bi' in the standard form, but since 'a' is 0 in this case, the standard form will be 'bi'. Therefore, the expression \(19i\sqrt{2}\) is already in its standard form.

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