Question

simplify complex numbers - worksheet

1. write both of the solutions to the following square root. give your answers as positive and negative multiples of ( i ).

( sqrt{-36} )

1. write both of the solutions to the following square root. give your answers as positive and negative multiples of ( i ).

( sqrt{-100} )

Explanation:

Problem 9:

Step1: Recall the imaginary unit definition

We know that \( i = \sqrt{-1} \), so we can rewrite \( \sqrt{-36} \) as \( \sqrt{36\times(-1)} \).
\[
\sqrt{-36}=\sqrt{36\times(-1)}
\]

Step2: Use square - root property

Using the property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (for \( a\geq0,b\geq0 \), here we extend it to the case of imaginary numbers), we have \( \sqrt{36\times(-1)}=\sqrt{36}\times\sqrt{-1} \). Since \( \sqrt{36} = 6 \) and \( \sqrt{-1}=i \), then \( \sqrt{-36}=6i \). But we need both positive and negative solutions. The square root of a number \( x \) has two solutions: \( \pm\sqrt{x} \). So the solutions for \( \sqrt{-36} \) are \( \pm 6i \).

Step1: Recall the imaginary unit definition

We know that \( i=\sqrt{-1} \), so we can rewrite \( \sqrt{-100} \) as \( \sqrt{100\times(-1)} \).
\[
\sqrt{-100}=\sqrt{100\times(-1)}
\]

Step2: Use square - root property

Using the property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (for \( a\geq0,b\geq0 \), here we extend it to the case of imaginary numbers), we have \( \sqrt{100\times(-1)}=\sqrt{100}\times\sqrt{-1} \). Since \( \sqrt{100} = 10 \) and \( \sqrt{-1}=i \), then \( \sqrt{-100}=10i \). But we need both positive and negative solutions. The square root of a number \( x \) has two solutions: \( \pm\sqrt{x} \). So the solutions for \( \sqrt{-100} \) are \( \pm 10i \).

Answer:

\( \pm 6i \)

Problem 10: