Questions
Questions

MATH136-12P-IO-202510-I-81X M04: Midterm Exam Review Quiz

Short answer

Simplify the complex numbers: 3+4i 2−i

View Explanation

View Explanation

Verified Answer
Please login to view
Step-by-Step Analysis
To simplify the given complex expression, we interpret the task as evaluating (3+4i) / (2 - i). First, multiply numerator and denominator by the complex conjugate of the denominator (2 + i) to rationalize: (3+4i)/(2 - i) * (2 + i)/(2 + i) = ( (3+4i)(2+i) ......Login to view full explanation

Log in for full answers

We've collected over 50,000 authentic exam questions and detailed explanations from around the globe. Log in now and get instant access to the answers!

Similar Questions

( 6 − 6 𝑖 − 9 3 − 27 𝑖 ) 256 = _ _ _ _ _ _ _ _ _ _   Hints: Convert the complex numbers to polar form. If   𝑧 = 𝑟 ( cos ⁡ 𝜃 + 𝑖 sin ⁡ 𝜃 ) then   𝑧 𝑛 = 𝑟 𝑛 ( cos ⁡ 𝑛 𝜃 + 𝑖 sin ⁡ 𝑛 𝜃 ) . If 𝑧 1 = 𝑟 1 ( cos ⁡ 𝜃 1 + 𝑖 sin ⁡ 𝜃 1 )   and   𝑧 2 = 𝑟 2 ( cos ⁡ 𝜃 2 + 𝑖 sin ⁡ 𝜃 2 ) then:           𝑧 1 𝑧 2 = 𝑟 1 𝑟 2 [ cos ⁡ ( 𝜃 1 + 𝜃 2 ) + 𝑖 sin ⁡ ( 𝜃 1 + 𝜃 2 ) ]   and   𝑧 1 𝑧 2 = 𝑟 1 𝑟 2 [ cos ⁡ ( 𝜃 1 − 𝜃 2 ) + 𝑖 sin ⁡ ( 𝜃 1 − 𝜃 2 ) ] .

The polar form of   𝑧 = − 6 7 + 3 2 7 𝑖 is: Hint: 𝑧 = 𝑎 + 𝑏 𝑖 = 𝑟 ( cos ⁡ 𝜃 + 𝑖 sin ⁡ 𝜃 ) where  𝑟 = 𝑎 2 + 𝑏 2   and   𝜃 = tan − 1 ⁡ 𝑏 𝑎 .          Also, don't forget to plot the complex number on the Argand diagram.

Given four complex numbers   𝑧 1 = 2 + 3 𝑖 , 𝑧 2 = − 7 − 5 𝑖 , 𝑧 3 = − 9 + 7 𝑖 , 𝑧 4 = 2 + 5 𝑖 . Calculate   | 𝑧 3 ¯ 𝑧 1 + 𝑧 2 𝑧 4 | . Formulae: If  𝑧 = 𝑎 + 𝑏 𝑖   then  𝑧 ¯ = 𝑎 − 𝑏 𝑖   and   | 𝑧 | = 𝑎 2 + 𝑏 2 . 𝑎 + 𝑏 𝑖 𝑐 + 𝑑 𝑖 = ( 𝑎 + 𝑏 𝑖 ) ( 𝑐 − 𝑑 𝑖 ) ( 𝑐 + 𝑑 𝑖 ) ( 𝑐 − 𝑑 𝑖 ) = ⋯

The figure shows the Argand diagram together with the complex number \(z\). If \(d=13\) and the imaginary part of \(z\) is \(-5\), find \(z\).

More Practical Tools for Students Powered by AI Study Helper

Join us and instantly unlock extensive past papers & exclusive solutions to get a head start on your studies!