Problem 18 - Precalculus Student Solutions Manual 5th

Question

Write each number as the product of a real number and i. $$\sqrt{-36}$$

Step-by-Step Solution

Answer

\(\sqrt{-36} = 6i\)

Step 1: Understand the problem

We need to express the number \(\texttt{\sqrt\texttt{-36}}\texttt{\sqrt{-36}}\) as the product of a real number and \(\texttt{\mathcal{i}}\texttt{+=i}\). \textttThe problem involves imaginary numbers, where \(i = \texttt{\sqrt{-1}}\texttt{\sqrt{-1}}\). So, we will use the property of imaginary numbers to solve this.

Step 2: Decompose the square root

Start by expressing \(\sqrt{-36}\) in terms of \(\sqrt{-1}\) and \(\sqrt{36}\). \(\sqrt{-36} = \sqrt{36 \times -1}\).

Step 3: Use properties of square roots

Apply the property of square roots: \(\texttt{\sqrt{\texttt{36}}}\) to \(\texttt{\sqrt{36}\=36 =}6\) and \(\texttt{\textttsqrt{\texttt{-1}}}\) to \(i\). Therefore, \(\sqrt{-36} = \sqrt{36} \times \sqrt{-1}\ =6 \times i\).

Step 4: Simplify the expression

Combine the results from the previous step to get \(\sqrt{-36}\) as \(6i\).

Key Concepts

complex numbers

Complex numbers might seem intimidating, but they serve as a powerful tool for solving various mathematical problems. A complex number is expressed in the form \(a + bi\), where:

• \texttt{a}\texttt{, the real part, is a real number.}\texttt{, the real part, is a real number.}
• \texttt{b}\texttt{ is also a real number and }\texttt{i}\texttt{ is the imaginary unit, which is defined as }\texttt{\texttt{\texttt{\texttt{\texttt{\texttt{\texttt{\texttt{\texttt{\texttt{\texttt{i \texttt{\texttt{the \texttt{\texttt{\texttt{\texttt{\texttt{\texttt{\texttt{\texttt{\texttt{\texttt{the \texttt{\texttt{\texttt{\texttt{the \texttt{\texttt{\texttt{\texttt{\texttt{the \texttt{\texttt{\texttt{\texttt{i}\texttt textttcttt to \texttt{\texttt{\texttexttt\texttexttt as as powtexttt \texttextt as asa as pow ass powt pow\textd assan textan powpow powasa power of pow\text i = is powasa powas pow pow pow pow pow powas powasa powas pow pow pow powas powasa powas pow pow pow pow powas powasa powas pow pow pow pow pow powas powasa powas pow pow pow pow pow pow pow pow pow pow pow pow pow pow pow pow pow pow pow pow powas pow pow pow