In Exercises 48–57, perform the indicated operations and write the result in standard form. √ (-32) - √ (-18)

Verified step by step guidance

1. insert step 1> Start by recognizing that you are dealing with square roots of negative numbers, which involve imaginary numbers. Recall that $\sqrt{-1}=i$, where $i$ is the imaginary unit.
2. insert step 2> Rewrite each square root in terms of $i$. For $\sqrt{-32}$, express it as $\sqrt{32}\times \sqrt{-1}=\sqrt{32}\times i$. Similarly, for $\sqrt{-18}$, express it as $\sqrt{18}\times \sqrt{-1}=\sqrt{18}\times i$.
3. insert step 3> Simplify $\sqrt{32}$ and $\sqrt{18}$ by factoring out perfect squares. $\sqrt{32}=\sqrt{16\times 2}=\sqrt{16}\times \sqrt{2}=4\sqrt{2}$. Similarly, $\sqrt{18}=\sqrt{9\times 2}=\sqrt{9}\times \sqrt{2}=3\sqrt{2}$.
4. insert step 4> Substitute the simplified forms back into the expressions: $\sqrt{-32}=4\sqrt{2}i$ and $\sqrt{-18}=3\sqrt{2}i$.
5. insert step 5> Perform the subtraction: $4\sqrt{2}i-3\sqrt{2}i=(4-3)\sqrt{2}i=\sqrt{2}i$.

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. In this problem, the square roots of negative numbers will yield complex results.

Square Roots of Negative Numbers

The square root of a negative number involves the imaginary unit 'i'. For example, √(-32) can be rewritten as √(32) * √(-1) = 4√2 * i. Understanding how to manipulate square roots of negative numbers is essential for solving problems involving complex numbers.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where 'a' and 'b' are real numbers. When performing operations with complex numbers, such as addition or subtraction, it is important to combine like terms to express the final result in this standard form.