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Chapter 1: Problem 18

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

Short Answer

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

Step by step solution

01

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.
02

Decompose the square root

Start by expressing \(\sqrt{-36}\) in terms of \(\sqrt{-1}\) and \(\sqrt{36}\). \(\sqrt{-36} = \sqrt{36 \times -1}\).
03

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, \(\sqrt{-36} = \sqrt{36} \times \sqrt{-1}\ =6 \times i\).
04

Simplify the expression

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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

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Most popular questions from this chapter

Simplify each power of i. $$i^{29}$$

Simplify each power of i. $$i^{27}$$

Show that \(-3+4 i\) is a solution of the equation \(x^{2}+6 x+25=0\)

Solve each rational inequality. Write each solution set in interval notation. $$\frac{x+3}{x-5} \leq 1$$$

Find each quotient. Write the answer in standard form \(a+b i .\) $$\frac{2}{3 i}$$
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